Primes in arithmetic progressions to large moduli I: Fixed residue classes
aa r X i v : . [ m a t h . N T ] J un PRIMES IN ARITHMETIC PROGRESSIONS TO LARGEMODULI I: FIXED RESIDUE CLASSES
JAMES MAYNARD
Abstract.
We prove new mean value theorems for primes in arithmetic pro-gressions to moduli larger than x / . Our main result shows that the primesare equidistributed for a fixed residue class over all moduli of size x / δ witha ‘convenient sized’ factor. As a consequence, the expected asymptotic holdsfor all but O ( δQ ) moduli q ∼ Q = x / δ and we get results for moduli aslarge as x / .Our proof extends previous techniques of Bombieri, Fouvry, Friedlanderand Iwaniec by incorporating new ideas inspired by amplification methods.We combine these with techniques of Zhang and Polymath tailored to our ap-plication. In particular, we ultimately rely on exponential sum bounds comingfrom the spectral theory of automorphic forms (the Kuznetsov trace formula)or from algebraic geometry (Weil and Deligne style estimates). Contents
1. Introduction 22. Acknowledgements 53. Outline 64. Corollaries 1.2, 1.3 and 1.4 from Theorem 1.1 105. Paper structure 126. Notation 147. Sieve decomposition 158. Type II estimates 199. Estimates for numbers with 4, 5 or 6 prime factors 2710. Sieve asymptotics 2911. Estimates for numbers with 3 prime factors 3312. Small factor Type II estimate 3713. Preparatory lemmas 4114. Dispersion estimates 4615. Fouvry-style estimates near x / x / x / x / Introduction
The Siegel–Walfisz theorem states that uniformly for q ≤ (log x ) A and ( a, q ) = 1(1.1) π ( x ; q, a ) = (cid:16) O A (cid:16) x ) A (cid:17)(cid:17) π ( x ) φ ( q ) . Without progress on the notorious problem of Siegel zeros we cannot hope to un-conditionally improve the range of q , despite it being very desirable to do so. Underthe assumption of the Generalized Riemann Hypothesis, however, the range when(1.1) holds can be extended to q ≤ x / / (log x ) B for B = B ( A ) sufficiently large interms of A . Moreover, it is conjectured [31] that (1.1) should hold for all q ≤ x − ǫ .For the purpose of many applications, an adequate substitute for the GeneralizedRiemann Hypothesis is the Bombieri–Vinogradov Theorem [2, 34]. This states thatfor every A > B = B ( A ) sufficiently large in terms of A we have(1.2) X q ≤ x / / (log x ) B sup ( a,q )=1 (cid:12)(cid:12)(cid:12) π ( x ; q, a ) − π ( x ) φ ( q ) (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A . If Q ≤ x / / (log x ) B then this implies that, for all but O A ( Q/ (log x ) A ) moduli q ≤ Q , we have (1.1) for all ( a, q ) = 1. In particular, ‘most’ moduli q satisfy(1.1), even when the moduli are essentially as large as can be handled with theGeneralized Riemann Hypothesis.As with the individual estimate (1.1), it is conjectured [12] that the range of q in(1.2) can be extended to q ≤ x − ǫ . However, it remains an important outstandingproblem in analytic number theory just to extend the range of moduli in (1.2) to q ≤ x / ǫ , thereby going ‘beyond the square-root barrier’ and producing estimateswhich are not directly implied by the Generalized Riemann Hypothesis. By expand-ing the summand via Dirichlet characters and the explicit formula, we see that thiswould imply some cancellation for sums over zeroes of different L -functions.In a series of papers [14, 16, 15, 13, 17, 18, 3, 4, 5] both separately and in col-laboration Bombieri, Fouvry, Friedlander and Iwaniec produced weakened variantsof (1.2) which were valid for moduli larger than x / . More recently, the key es-timate in the work of Zhang [35] on bounded gaps between primes was a variantof (1.2) valid for moduli slightly larger than x / provided the moduli only hadsmall prime factors. Zhang’s work was extended by the Polymath project [32], andrelated recent results were obtained by Fouvry and Radziwi l l [21, 22], Drappeau[10], Drappeau, Radziwi l l and Pratt [11] and Assing, Blomer and Li [1].In this paper we also consider variants of (1.2) which are valid for moduli beyond x / , extending some of the previous estimates, particularly those of Bombieri,Friedlander and Iwaniec [3, 4, 5]. As with these previous works, we will makeuse of estimates of Deshouillers–Iwaniec [9] for sums of Kloosterman sums comingfrom the Kuznetsov trace formula which requires us to only consider the situationwhen a is fixed (or a small power of x ). Crucially we refine some estimates via anamplification method, allowing us to handle several of the critical cases which werepreviously inaccessible. We combine these ideas with ideas originating in the workZhang [35] and Polymath [32] based on exponential sum estimates from algebraicgeometry which also require some divisibility conditions on the moduli. Together, RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 3 our methods enable us to prove qualitatively new variants of (1.2) for larger modulias well as giving quantitative improvements.Our main result proves an extension of (1.2) for moduli which have a convenientlysized divisor.
Theorem 1.1.
Let a ∈ Z , let ǫ > and let Q , Q satisfy Q Q < x − ǫ , (1.3) Q Q < x − ǫ , (1.4) Q Q < x − ǫ . (1.5) Then for every
A > we have X q ≤ Q ( q ,a )=1 X q ≤ Q ( q ,a )=1 (cid:12)(cid:12)(cid:12) π ( x ; q q , a ) − π ( x ) φ ( q q ) (cid:12)(cid:12)(cid:12) ≪ a,ǫ,A x (log x ) A . The main qualitative feature of Theorem 1.1 is that it bounds the error term forprimes in arithmetic progressions with absolute values and a strong error term,whilst simultaneously applying to ‘most’ moduli.The conditions on Q , Q should be thought of as restricting to moduli q < x / with a ‘convenient sized factor’, where the restriction on the possible size of thisfactor is weak when considering moduli of size x / δ with δ small, but more re-strictive as δ grows. In particular, setting Q Q = x / δ and allowing Q to vary,we see that Theorem 1.1 can be rephrased as follows. Corollary 1.2.
Let a ∈ Z , < δ < / , and < η < (1 − δ ) / and Q δ,η := n q ≤ x / δ : ∃ d | q s.t. x δ + η < d < min (cid:16) x / x δ/ η , x / x δ + η (cid:17)o . Then for every
A > we have X q ∈Q δ,η ( q,a )=1 (cid:12)(cid:12)(cid:12) π ( x ; q, a ) − π ( x ) φ ( q ) (cid:12)(cid:12)(cid:12) ≪ a,η,A x (log x ) A . One should think of η as a small constant, so Q δ essentially counts moduli q of size x / δ which have a ‘conveniently sized factor’ in the interval [ x δ , x / − δ/ ].When δ is small ‘most’ moduli of size x / δ have a divisor in the range [ x δ , x / ],and so most moduli are contained in Q δ,η . In particular, we can extend the asymp-totic (1.1) to ‘most’ moduli of size x / δ . This is made precise by the followingresult. Corollary 1.3.
Let a ∈ Z , < δ < / , A > and Q ≤ x / δ . Then for all butat most δQφ ( a ) /a moduli q ∈ [ Q, Q ] with ( q, a ) = 1 we have π ( x ; q, a ) = (cid:16) O a,δ,A (cid:16) x ) A (cid:17)(cid:17) π ( x ) φ ( q ) . For example, if Q ≤ x / / , we see that 99% of moduli q ∈ [ Q, Q ] with( a, q ) = 1 have the expected asymptotic count for the number of primes. JAMES MAYNARD
When Q ≈ x / , Theorem 1.1 allows us to obtain non-trivial result for moduli aslarge as x / − ǫ ; this compares quite favourably with previous results. Explicitly,we have the following variant. Corollary 1.4.
Let a ∈ Z , ǫ > . Then we have for every A > X q ≤ x / ( q ,a )=1 X q ≤ x / − ǫ ( q ,a )=1 (cid:12)(cid:12)(cid:12) π ( x ; q q , a ) − π ( x ) φ ( q q ) (cid:12)(cid:12)(cid:12) ≪ a,ǫ,A x (log x ) A . Comparison with previous results.
The key qualitative features of Theo-rem 1.1 are that it gives good savings over the trivial bound for ‘most’ moduli ofsize x / δ and treats the error terms with absolute values. This is the first suchresult.We first record three key results for comparison. The first is a combination of themain results of [4] and [5]. Theorem A (Bombieri, Friedlander, Iwaniec) . Let a ∈ Z , δ > and Q = x / δ .Then we have X q ≤ Q ( q,a )=1 (cid:12)(cid:12)(cid:12) π ( x ; q, a ) − π ( x ) φ ( q ) (cid:12)(cid:12)(cid:12) ≪ a δ x log x + x (log log x ) O (1) (log x ) . The second is [3, Theorem 10].
Theorem B (Bombieri, Friedlander, Iwaniec) . Let
A > and λ d be a well-factorable sequence of level Q < x / − ǫ . Then we have X q ≤ Q ( q,a )=1 λ q (cid:16) π ( x ; q, a ) − π ( x ) φ ( q ) (cid:17) ≪ a,A,ǫ x (log x ) A . The final result is Polymath’s refinement [32] of Zhang’s work [35].
Theorem C (Zhang, Polymath) . Let a ∈ Z , < δ < / and Q = x / δ . Thenthere is a constant η = η ( δ ) > such that for every A > X q ≤ Q ( q,a )=1 p | q ⇒ p ≤ x η (cid:12)(cid:12)(cid:12) π ( x ; q, a ) − π ( x ) φ ( q ) (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A . Theorem A is non-trivial only when δ is small, and wins only a small amount overthe trivial bound, but involves all moduli of size Q . By comparison, Theorem 1.1gives a non-trivial result for significantly larger moduli, with better savings, butonly holds for a subset of moduli. When δ is small, ‘most’ moduli are included, andso it automatically implies a result of the sort of Theorem A. Indeed, the methodof proof is an extension and refinement of the ideas underlying Theorem A, andthe proof contains the proof of Theorem A as a special case. (One should be ableto obtain a version of Theorem A with δ replaced by δ , for example.) RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 5
Theorem B has the advantage over Theorem 1.1 that it can handle significantlylarger moduli (of size x / − ǫ compared with x / − ǫ ). However, Theorem 1.1 hasthe advantage that it applies to a much larger set of moduli (the moduli appearingin Theorem B must have many flexible factorizations, which is a small proportionof q ≤ Q ), and it has the advantage that the error term is handled with absolutevalues rather than a well-factorable weight. In the second paper of this series wewill build upon ideas in this paper to produce a variant of Theorem B which canhandle larger moduli than x / .Theorem C has a distinct advantage over Theorem 1.1 in that it is uniform withrespect to the residue class a . It only applies to special moduli, however, whichare quite rare, whereas Theorem 1.1 holds for a fairly large proportion of moduli(depending on the size of δ ). Moreover, Theorem 1.1 holds for moduli as largeas x / − ǫ , which is slightly larger than the largest moduli covered by TheoremC (of size x / − ǫ ). Indeed, for a fixed residue class, Theorem 1.1 implies thatTheorem C holds with moduli of size up to x / − ǫ . In the third paper of thisseries we will produce an estimate which is completely uniform with respect to theresidue class, as in (1.2). 2. Acknowledgements
I would like to thank Etienne Fouvry, John Friedlander, Andrew Granville, BenGreen, Henryk Iwaniec, Kyle Pratt and Maksym Radziwi l l for useful discussionsand suggestions. JM is supported by a Royal Society Wolfson Merit Award, andthis project has received funding from the European Research Council (ERC) un-der the European Unions Horizon 2020 research and innovation programme (grantagreement No 851318).
JAMES MAYNARD Outline
High-level overview.
The overall structure of our argument follows all previ-ous results; we use a combinatorial decomposition of the primes (such as Vaughan’sidentity or Heath-Brown’s identity) to transform the problem into that of estimat-ing various convolutions of sequences in arithmetic progressions. We then producevarious different estimates depending on the sizes of the factors in the convolu-tion. In each case we use a version of the Linnik dispersion method and Fourierexpansion to reduce the problem to estimating complicated exponential sums. Af-ter some significant manipulation, these exponential sums are then estimated usingresults based on ideas from algebraic geometry (the Weil bound or Deligne-style es-timates) or from automorphic forms (Deshouillers–Iwaniec-style estimates for sumsof Kloosterman sums via the Kuznetsov trace formula).On a basic level, this paper refines ideas of Bombieri–Friedlander–Iwaniec and com-bines them with those of Zhang to produce a hybrid result. Indeed, Zhang’s workparticularly made use of the flexible factorizations of the moduli, and we produce ahybrid result which requires some, but weaker, divisibility conditions for our moduliby combining it with previous ideas coming from ‘Kloostermania’ estimates.Unfortunately, na¨ıvely combining the two methods produces very unsatisfactoryresults, since the ‘worst-case’ is the same for both approaches. To overcome thisissue it is vital that we refine some of the estimates of Bombieri–Friedlander–Iwaniecto handle what was previously the hardest case. This refinement isn’t quite enoughto cover all cases on its own, but after this refinement the new worst case scenario forthe Bombieri–Friedlander–Iwaniec-style estimates is one of the best case scenariosfor the Zhang-style estimates, and so the methods combine effectively.3.2.
Critical case in the Bombieri–Friedlander–Iwaniec argument.
Let usrecall the situation in the proof of [5], where we consider primes in arithmeticprogressions to moduli of size Q = x / δ with δ > x / O ( δ ) .(2) Convolutions of 4 factors each supported on numbers of size x / O ( δ ) .(This is actually an over-simplification as there are other related bad cases, such as4 factors with one of size x / O ( δ ) and three of size x / O ( δ ) , but the cases aboveare the key ones. The proof in [5] also gives a wider variety of bad cases, but asmall variation on the techniques can handle these other cases without new ideas.)In the bad cases above all factors are restricted to relatively short ranges on alogarithmic scale, and so one can simply discard the contributions from these terms.This produces an error term of size roughly δ times the trivial bound, which thenautomatically gives Theorem A.Zhang’s work (and its refinements by Polymath) give a method to cover any givencombination of convolution lengths, provided the moduli under consideration fac-torize in a suitable manner. Unfortunately the ‘hardest’ situation is dealing with RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 7 convolutions of five factors each of length x / . The argument for this case wouldproduce a result with severe constraints on the factorizations of the moduli, andwould almost degenerate to Theorem C, producing little qualitatively (or quanti-tatively new).3.3. The original BFI argument.
Let us quickly recall why the argument thatBombieri–Friedlander–Iwaniec use for handling convolutions of five factors fails.Grouping into variables n ∼ N , n ∼ N and n ∼ N with N , N ≈ x / and N ≈ x / , one wants to estimate(3.1) X q ∼ Q c q X n ∼ N α n X n ∼ N β n X n ∼ N (cid:16) n n n ≡ a (mod q ) − ( n n n ,q )=1 φ ( q ) (cid:17) (for suitable divisor-bounded coefficients c q , α n , β n .) Applying Cauchy’s inequal-ity in n and q the key term one needs to estimate is (a smoothed variant of)(3.2) X q ∼ Q X n ,n ′ ∼ N X n ,n ′ ∼ N n n ≡ n ′ n ′ (mod q ) β n β n ′ X n ∼ N n n n ≡ a (mod q ) . Fourier completing the inner sum, using Bezout’s identity to flip the moduli in theresulting exponential and substituting qr = n n − n ′ n ′ leaves us to estimate asum like(3.3) X r ∼ N N /Q X n ,n ′ ∼ N X n ,n ′ ∼ N n n ≡ n ′ n ′ (mod r ) β n β n ′ X h ∼ Q/N e (cid:16) ahn ′ n ′ n n (cid:17) . The approach of Bombieri–Friedlander–Iwaniec is then to apply Cauchy’s inequalityto r, n , n , n ′ and combine n , n together, leading one to bound(3.4) X r ∼ N N /Q X n ∼ N N X n ′ ∼ N (cid:12)(cid:12)(cid:12) X h ∼ Q/N X n ′ ∼ N n ′ n ′ ≡ n (mod r ) β n ′ e (cid:16) ahn ′ n ′ n (cid:17)(cid:12)(cid:12)(cid:12) . Unfortunately the diagonal terms are acceptable only if N /r > (log x ) A Q/N (sothat the the savings outweigh the losses from completion of sums), which requiresthat N > (log x ) A N . This condition (just) fails if N ≈ N , and this means thatthis approach cannot handle the case of 5 equally sized factors. It doesn’t seem likeany other rearrangement of the sums can handle this situation either.3.4. A modified argument.
To address the issue raised above we use an ideainspired by the ‘amplification’ method of Friedlander–Iwaniec (see [24]). Normallythe amplification method enables one to obtain a non-trivial bound by consideringa larger average over a family, but amplifying the contribution from the sum ofinterest, which increases the size of the diagonal terms. In our situation we havean almost opposite situation - we wish to reduce (or ‘de-amplify’) the contributionfrom the diagonal terms. We do this by first artificially introducing a congruencecondition n n ≡ b (mod c ) which we average over at the outset, so we consider in JAMES MAYNARD place of (3.1)1 C X c ∼ C X b (mod c ) X q ∼ Q c q X n ∼ N α n X n ∼ N β n X n ∼ N n n ≡ b (mod c ) ∆ , where ∆ is the same inner sum as before. We now follow the previous approach,applying Cauchy in c, b, q and n , leaving us to estimate the equivalent of (3.2) X c ∼ C X q ∼ Q X n ,n ′ ∼ N X n ,n ′ ∼ N n n ≡ n ′ n ′ (mod qc ) β n β n ′ X n ∼ N n n n ≡ a (mod q ) . We can Fourier complete the inner sum and apply Bezout as before (with no ad-ditional losses) but now our substitution becomes qr ′ c = n n − n ′ n ′ , and theequivalent of (3.3) becomes X r ′ ∼ N N /QC X c ∼ C X n ,n ′ ∼ N X n ,n ′ ∼ N n n ≡ n ′ n ′ (mod r ′ c ) β n β n ′ X h ∼ Q/N e (cid:16) ahn ′ n ′ n n (cid:17) . This has essentially forced the r variable to have a factor c ∼ C compared withwhat we had before, and we can choose any C ∈ [1 , N N /Q ]. Thinking of C = x ǫ ,we can then apply Cauchy in r, n , n , n ′ , leaving us to estimate X r ∼ N N /QC X n ∼ N N X n ′ ∼ N (cid:12)(cid:12)(cid:12)X c ∼ C X h ∼ Q/N X n ′ ∼ N n ′ n ′ ≡ n (mod rc ) β n ′ e (cid:16) ahn ′ n ′ n n (cid:17)(cid:12)(cid:12)(cid:12) . This has effectively reduced the size of the diagonal terms by a factor of C , so theyare now admissible. The off-diagonal terms will be larger by a factor polynomial in C , which is acceptable since for C a small enough power of x since we had a power-saving bound beforehand. This therefore allows us to treat the critical situation offive factors of size x / .To make this argument work we rely on progress towards the Selberg eigenvalueconjecture for Maass forms, and in particular we use the work of Kim–Sarnak [29]coming from automorphy of symmetric fourth power L -functions in the estimatesof Deshouillers–Iwaniec [9] on sums of Kloosterman sums.3.5. Combining with ideas of Zhang.
Unfortunately, the estimates of Bombieri–Friedlander–Iwaniec type are still not useful for handling products of 4 factors ofsize x / O ( δ ) . To handle these terms we use ideas based on Zhang’s work, and itis these ideas which require us to assume that our moduli have a convenient sizedprime factor (if we could handle these quadruple convolutions for all q , then wecould get a corresponding version of Theorem 1.1 valid for all q ∼ x / δ ).One can combine two of the factors together to get a factor very close to x / .Provided the moduli factorize such that one factor is a small power of x and theother is close to x / , the Zhang-style estimates can cover these cases, and moreoverit is one of the most efficient parts of Zhang’s argument. This means for moduliwith a ‘convenient sized’ prime factor, we can handle these terms. RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 9
Together with our previous refinement, this enables us to handle all the terms nothandled by [5], and so would give a variant of Theorem 1.1 when δ is sufficientlysmall.3.6. Quantitative bounds.
To get a good quantitative estimate, we refine severaldifferent auxiliary estimates in other regimes, making use of the fact that we assumethat our moduli have a conveniently sized divisor to enhance the previous estimates.In particular, we refine work of Fouvry [14] for estimates with a factor close to x / ,refine work of Zhang [35] and Polymath [32] for estimates with a factor close to x / ,produce a new small divisor estimate for estimates with a factor close to x / , andrefine estimates on triple convolutions of smooth sequences by producing estimatestailored to our situation. These refinements enable us to handle moduli as large as x / . Until the recent work of Fouvry-Kowalski-Michel [19] and Polymath [32],the much easier problem of showing that triple divisor function d ( n ) satisfied aBombieri-Vinogradov type theorem to moduli larger than x / was not known, forexample. We have also swept several additional technical details under the carpetto emphasize to the reader what we view as the most important new ideas. Corollaries 1.2, 1.3 and 1.4 from Theorem 1.1
Before properly embarking on the proof of Theorem 1.1, we first show how Corol-laries 1.2-1.4 follow from Theorem 1.1.
Proof of Corollary 1.2 assuming Theorem 1.1.
For notational convenience set D := x δ + η / D := min( x / − δ/ − η , x / − δ − η ). By considering the d in dyadicranges we see that Q δ,η ⊆ [ D =2 j D ≤ D ≤ D n q ≤ x / δ : ∃ d | q s.t. d ∈ [ D, D ] o = [ D =2 j D ≤ D ≤ D n q q ≤ x / δ : q ∈ [ D, D ] o . We now set Q := 2 D and Q := x / δ /D , noting that if q q ≤ x / δ and q ∈ [ D, D ] then q ≤ Q and q ≤ Q . If D ≤ D ≤ D we have Q Q = 2 x δ D ≪ x − η ,Q Q = 2 D x / δ ≪ x − η ,Q Q = 2 Dx / δ ≪ x − η . Thus, by the union bound and applying Theorem 1.1 (with ǫ = η/ X q ∈Q δ,η ( q,a )=1 (cid:12)(cid:12)(cid:12) π ( x ; q, a ) − π ( x ) φ ( q ) (cid:12)(cid:12)(cid:12) ≪ (log x ) sup D ≤ D ≤ D X q ≤ D X q ≤ x / δ /D (cid:12)(cid:12)(cid:12) π ( x ; q q , a ) − π ( x ) φ ( q q ) (cid:12)(cid:12)(cid:12) ≪ a,A,η x (log x ) A . (cid:3) Proof of Corollary 1.3 assuming Theorem 1.1.
We let η = ǫδ . Since 0 < δ < / q ∼ Q with q / ∈ Q δ,η must be the product m m with m < x (2+ ǫ ) δ and P − ( m ) ≥ x / − (7 / ǫ ) δ ≥ ( Q/m ) / , where we use P − ( m ) to denote thesmallest prime factor of m . We recall that as t → ∞ X n ≤ tP − ( n ) ≥ t γ (cid:16) ω (cid:16) γ (cid:17) + o γ (1) (cid:17) tγ log t , where ω ( u ) is the Buchtab function, which is piecewise continuous and satisfies ω (7) < /
7. Thus we see that for ǫ small enough and x large enough X q ∈ [ Q, Q ] \Q δ,η ( q,a )=1 ≤ X m ≤ x (2+ ǫ ) δ ( m ,a )=1 X Q/m ≤ m ≤ Q/m P − ( m ) ≥ ( Q/m ) / X m ≤ x (2+ ǫ ) δ ( m ,a )=1 (7 ω (7) + o (1)) Qm log( Q/m ) ≤ φ ( a ) a (4 + o (1)) Q (2 + ǫ ) δ / − (2 + ǫ ) δ ≤ φ ( a ) δQa . RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 11
By Corollary 1.2, for all but O ( Q/ log A x ) moduli q ∈ Q δ,η with ( q, a ) = 1 we have π ( x ; q, a ) = π ( x ) φ ( q ) + O a,A,δ (cid:16) x (log x ) A (cid:17) . Putting this all together gives the result. (cid:3)
Proof of Corollary 1.4 assuming Theorem 1.1.
This is immediate from Theorem 1.1taking Q = x / , Q = x / − ǫ , and then redefining ǫ . (cid:3) Thus we are left to establish Theorem 1.1. Paper structure
The dependency diagram between the main propositions in the paper is as follows.
Prop. 7.4Prop. 7.3Prop. 7.2Prop. 7.1 Thm. 1.1Prop. 10.1Prop. 8.1Prop. 11.1Prop. 12.1Prop. 12.2Prop. 8.3Prop. 8.2
In the first half of the paper, Sections 7-12, we reduce the proof of Theorem 1.1 tothe task of establishing the technical Propositions 11.1, 12.1, 12.2, 8.2 and 8.3. Thesecond half of the paper, Sections 13-20 is then spent establishing these propositions.In Section 7, we reduce the proof of Theorem 1.1 to establishing Propositions 7.4,7.2, 7.1 and 7.3. This is done by a combinatorial sieve decomposition, relying onHarman’s sieve.In Section 8 we first reduce the proof of our Type II result (Proposition 7.1) tothe more general Proposition 8.1. We then reduce Proposition 8.1 to establishingthe more technical Proposition 8.2 and Proposition 8.3, which establish Type IIestimates for convolutions which are effective either close to, or further from thebalanced length case.In Section 9, we deduce Proposition 7.3 handling numbers with four or more primefactors from Proposition 8.3 and Proposition 8.1 by performing more combinatorialmanipulations.In Section 10 we establish Proposition 7.2 from Proposition 8.1 and the more tech-nical Proposition 10.1. These arguments are based on estimates associated withthe fundamental lemma of sieve methods and Harman’s sieve.In Section 11 we deduce Proposition 7.4 on numbers with three prime factors fromPropositions 7.3 and 10.1, and the more technical Proposition 11.1.In Section 12 we deduce Proposition 10.1 on convolutions with a short factor fromthe more technical Proposition 12.1 and 12.2.In Section 13 and Section 14 we establish various preparatory lemmas and a generalestimate for using the Linnik dispersion method to reduce the problems consideredto exponential sums.
RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 13
In Section 15 we establish Proposition 12.1, refining ideas of Fouvry [14].In Section 16, we establish Proposition 12.2, which is a new estimate similar toideas of Fouvry [14].In Section 17 we establish Proposition 8.2, which is a refinement of an estimate ofZhang [35].In Section 18 we establish Proposition 8.3, refining previous work of Bombieri,Friedlander and Iwaniec [4, Theorem 4].In Section 19, we establish a couple of preparatory lemmas for the later divisorfunction estimates of Section 20.In Section 20 we establish Proposition 11.1 on the triple divisor function, developingideas of Friedlander, Iwaniec [23], Heath-Brown [27], Fouvry, Kowalski, Michel [19]and Polymath [32]. Notation
We will use the Vinogradov ≪ and ≫ asymptotic notation, and the big oh O ( · )and o ( · ) asymptotic notation. f ≍ g will denote the conditions f ≪ g and g ≪ f both hold. Dependence on a parameter will be denoted by a subscript.We will view a (the residue class we count arithmetic functions in to different moduli q ) as a fixed positive integer throughout the paper, and any constants impliedby asymptotic notation will be allowed to depend on a from this point onwards.Similarly, throughout the paper, we will let ǫ be a single fixed small real number; ǫ = 10 − would probably suffice. Any bounds in our asymptotic notation willalso be allowed to depend on ǫ .The letter p will always be reserved to denote a prime number. We use φ todenote the Euler totient function, e ( x ) := e πix the complex exponential, τ k ( n ) the k -fold divisor function, µ ( n ) the M¨obius function. We let P − ( n ), P + ( n ) denotethe smallest and largest prime factors of n respectively, and b f denote the Fouriertransform of f over R - i.e. b f ( ξ ) = R ∞−∞ f ( t ) e ( − ξt ) dt . We use to denote theindicator function of a statement. For example, n ≡ a (mod q ) = ( , if n ≡ a (mod q ) , , otherwise . We will use ( a, b ) to denote gcd( a, b ) when it does not conflict with notation forordered pairs. For ( n, q ) = 1, we will use n to denote the inverse of the integer n modulo q ; the modulus will be clear from the context. For example, we maywrite e ( an/q ) - here n is interpreted as the integer m ∈ { , . . . , q − } such that mn ≡ q ). Occasionally we will also use λ to denote complex conjugation; thedistinction of the usage should be clear from the context. For a complex sequence α n ,...,n k , k α k will denote the ℓ norm k α k = ( P n ,...,n k | α n ,...,n k | ) / .Summations assumed to be over all positive integers unless noted otherwise. Weuse the notation n ∼ N to denote the conditions N < n ≤ N .We will let z := x / (log log x ) and y := x / log log x two parameters depending on x , which we will think of as a large quantity. We will let ψ : R → R denote afixed smooth function supported on [1 / , /
2] which is identically equal to 1 on theinterval [1 ,
2] and satisfies the derivative bounds k ψ ( j )0 k ∞ ≪ (4 j j !) for all j ≥ Definition 1 (Siegel-Walfisz condition) . We say that a complex sequence α n sat-isfies the Siegel-Walfisz condition if for every d ≥ , q ≥ and ( a, q ) = 1 andevery A > we have (6.1) (cid:12)(cid:12)(cid:12) X n ∼ Nn ≡ a (mod q )( n,d )=1 α n − φ ( q ) X n ∼ N ( n,dq )=1 α n (cid:12)(cid:12)(cid:12) ≪ A N τ ( d ) B (log N ) A . We note that α n certainly satisfies the Siegel-Walfisz condition if α n = 1, if α n = µ ( n ) or if α n is the indicator function of the primes. RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 15 Sieve decomposition
In this section we prove Theorem 1.1 assuming five more technical propositions,which we will then gradually establish over the rest of the paper. We do this byapplying a sieve decomposition to break the count of primes in arithmetic progres-sions into counts of integers with particular prime factorizations, which can thenbe estimated using the relevant proposition. The sieve decomposition is based onideas based on Harman’s sieve (see [25]), but we could have used the Heath-Brownidentity and some combinatorial lemmas as an alternative (we hope that this ar-rangement will be more useful for future work).Define S n and S d ( z ) (depending on integers a, q , q satisfying ( a, q q ) = 1 whichwe suppress from the notation for convenience) for integers n, d and a real z by S n := n ≡ a (mod q q ) − φ ( q q ) ( n,q q )=1 ,S d ( z ) := X n ∼ x/dP − ( n ) >z S dn . With this notation, we are able to state our key propositions.
Proposition 7.1 (Type II estimate) . Let Q , Q satisfy (1.3) and (1.4) . Let P , . . . , P J ≥ x / ǫ be such that P · · · P J ≪ x and x / ǫ ≤ Y j ∈J P j ≤ x / − ǫ for some subset J ⊆ { , . . . , J } . Then we have X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X ∗ p ,...,p J p i ∼ P i ∀ i S p ··· p J ( p J ) (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A . Here P ∗ indicates that the summation is restricted by O (1) inequalities of the form p α · · · p α J J ≤ B . The implied constant may depend on all such exponents α i , butnone of the quantities B . Proposition 7.2 (Sieve asymptotics) . Let
A > . Let x / ǫ ≥ P ≥ · · · ≥ P r ≥ x / ǫ be such that P · · · P r ≤ x / and such that either r = 1 or P r ≤ x / ǫ .Let Q , Q satisfy (1.3) , (1.4) and (1.5) .Then we have X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X ∗ p ,...,p r p i ∼ P i ∀ i S p ··· p r ( x / ǫ ) (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A . Here P ∗ means that the summation is restricted to O (1) inequalities of the form p α · · · p α r r ≤ B for some constants α , . . . α r . The implied constant may depend onall such exponents α i , but none of the quantities B .Moreover, we also have the related estimate X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) S ( x / ǫ ) (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A . Proposition 7.3 (Numbers with 4 or more prime factors) . Let
A > . Let J ≥ and P ≥ · · · ≥ P J ≥ x / ǫ with P · · · P J ≍ x . Let Q , Q satisfy (1.3) and (1.4) . Then we have X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X ∗ p ,...,p J p i ∼ P i ∀ i S p ··· p J (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A . Here P ∗ indicates that the summation is restricted by O (1) inequalities of the form p α · · · p α J J ≤ B . The implied constant may depend on all such exponents α i , butnone of the quantities B . Proposition 7.4 (Numbers with three prime factors) . Let
A > and let P , P , P ∈ [ x / , x / ǫ ] with P P P ≍ x . Then we have X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X ∗ p ,p ,p p i ∼ P i ∀ i S p p p (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A . Here P ∗ means that the summation is restricted to O (1) inequalities of the form p α p α p α ≤ B for some constants α , α , α . The implied constant may depend onall such exponents α i , but none of the quantities B . Proposition 7.1 will be established in Section 8, Proposition 7.2 will be establishedin Section 10, Proposition 7.3 will be established in Section 9, and Proposition 7.4will be established in Section 11.Our decomposition will rely on suitable applications of the following elementaryidentity.
Lemma 7.5 (Buchstab’s identity) . Let z < z . Then S d ( z ) = S d ( z ) − X z
This is inclusion-exclusion based on the smallest prime factor of n in thesum in S d ( z ). (cid:3) We are now in a position to prove Theorem 1.1.
Proof of Theorem 1.1 assuming Propositions 7.1, 7.2, 7.3 and 7.4.
We first estab-lish some notation. Let z , z , z be defined in terms of x by z := x / ǫ , z := x / ǫ , z := x / − ǫ . For the purposes of this section (but this section only), we assume that the symbols r, s, t, u, v represent prime numbers, in the same manner that we use the symbol p .Our goal is to estimate(7.1) X q ∼ Q X q ∼ Q ( q q ,a )=1 | S (2 x / ) | = X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X p ∼ xp ≡ a (mod q q ) − φ ( q ) X p ∼ x ( p,q q )=1 (cid:12)(cid:12)(cid:12) RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 17 by decomposing S (2 x / ) into terms which can be handled by one of Proposition7.1, Proposition 7.2, Proposition 7.3 or Proposition 7.4. By Buchstab’s identity S (2 x / ) = S ( z ) − X z
x / S pr ( r ) . (7.3)(We have simplified some of the conditions of summation above.)By Proposition 7.2, the first term in (7.3) contributes negligibly to (7.1). By Propo-sition 7.1, the third term also contributes negligibly. Thus we are left to considerthe remaining three terms of (7.3).We first consider the final term of (7.3). Since pr > x this counts products ofexactly three primes. Specifically X z RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 19 Type II estimates In this section we establish our Type II estimate Proposition 7.1 via a slight variant,namely Proposition 8.1, given below. We do this assuming two technical proposi-tions, Propositions 8.2 and Proposition 8.3 along with the preexisting results givenby Lemmas 8.4 8.5. Proposition 8.1 (Type II estimate) . Let A > and let Q , Q satisfy (1.3) and (1.4) . Let P , . . . , P J ≥ x / ǫ be such that P · · · P J ≍ x and x / ǫ ≤ Y j ∈J P j ≤ x / − ǫ for some subset J ⊆ { , . . . , J } . Then we have X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X ∗ p ,...,p J p i ∼ P i ∀ i S p ··· p J (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A . Here P ∗ indicates that the summation is restricted by O (1) inequalities of the form p α · · · p α J J ≤ B . The implied constant may depend on all such exponents α i , butnone of the quantities B . We recall that here (and throughout the paper) S n is defined by S n := n ≡ a (mod q q ) − φ ( q q ) ( n,q q )=1 . We begin by noting that Proposition 7.1 follows immediately from Proposition 8.1. Proof of Proposition 7.1 assuming Proposition 8.1. We note that S p ··· p J ( p J ) is aweighted sum over integers p · · · p J n ∼ x with P − ( p . . . p J n ) ≥ x / ǫ , and sowith at most 6 prime factors. Expanding this into separate terms according to theexact number of prime factors, we see that each term is of the form considered byProposition 8.1, and so the result follows immediately. (cid:3) Thus we wish to establish Proposition 8.1. Our first key estimate is essentially anestimate in the work of Zhang [35], and will be established in Section 17. Proposition 8.2 (Zhang-style estimate) . Let A > . Let N, M, Q , Q ≥ with N M ≍ x be such that Q Q < x − ǫ , x ǫ Q < N < x − ǫ Q . Let β m , α n be complex sequences such that | α n | , | β n | ≤ τ ( n ) B and such that α n satisfies the Siegel-Walfisz condition (6.1) and α n is supported on n with all primefactors bigger than z = x / (log log x ) . Let ∆( q ) := X m ∼ M X n ∼ N α n β m (cid:16) mn ≡ a (mod q ) − ( mn,q )=1 φ ( q ) (cid:17) . Then we have X q ∼ Q X q ∼ Q ( q q ,a )=1 | ∆( q q ) | ≪ A,B x log A x . Our next key proposition is a refinement of work of Bombieri, Friedlander andIwaniec [4, Theorem 4], and will be established in Section 18. Proposition 8.3 (New estimate for triple convolutions) . Let A, B > , KLM ≍ x , min( K, L, M ) > x ǫ , a = 0 and x / − ǫ > Q > x / (log x ) − A . Let L, K satisfy Qx ǫ < KL,QK < x − ǫ ,Q K L < x − ǫ ,Q K L < x − ǫ . Let η k , λ ℓ , β m be complex sequences such that | η n | , | λ n | , | β n | ≤ τ ( n ) B and such that η k satisfies the Siegel-Walfisz condition (6.1) , and such that η k , λ ℓ be supported onintegers with all prime factors bigger than z . Let ∆ B ( q ) := X k ∼ K X ℓ ∼ L X m ∼ M η k λ ℓ α m (cid:16) kℓm ≡ a (mod q ) − ( kℓm,q )=1 φ ( q ) (cid:17) . Then we have X q ∼ Q ( q,a )=1 | ∆ B ( q ) | ≪ A,B x (log x ) A . We also require two results of Bombieri, Friedlander and Iwaniec. Lemma 8.4 (Triple convolution of rough sequences) . Let A, B > , KLM ≍ x , min( K, L, M ) > x ǫ , a = 0 and x / − ǫ > Q > x / (log x ) − A . Let L, K satisfy Qx ǫ < KL, K L < x / − ǫ , K L < x − ǫ , K L < x − ǫ . Let η k , λ ℓ , β m and ∆ B ( q ) be as in Proposition 8.3. Then we have X q ∼ Q ( q,a )=1 | ∆ B ( q ) | ≪ A,B x (log x ) A . Proof. This is [4, Theorem 3] (with the roles of K and L reversed). (cid:3) Lemma 8.5 (Triple convolution involving a smooth sequence) . Let A, B > , KLM ≍ x with min( K, L, M ) > x ǫ and let β m , η k be complex sequences with | β n | , | η n | ≤ τ ( n ) B . Let I L ⊆ [ L, L ] be an interval. Let L, K, M satisfy Qx ǫ < KL, K Q < Lx − ǫ , KQ < Lx − ǫ . Then we have X q ∼ Q ( q,a )=1 (cid:12)(cid:12)(cid:12) X k ∼ K X m ∼ M X ℓ ∈I L P − ( ℓ ) ≥ z η k β m (cid:16) kmℓ ≡ a (mod q ) − ( kℓm,q )=1 φ ( q ) (cid:17)(cid:12)(cid:12)(cid:12) ≪ A,B x (log x ) A . Proof. This is [4, Theorem 5*]. (cid:3) Combining Proposition 8.3 with Lemma 8.4 gives the following. RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 21 Lemma 8.6. Let A, B > , KLM ≍ x , min( K, L, M ) > x ǫ , a = 0 and Q >x / − ǫ/ . Let L, K satisfy Qx ǫ ≤ KL ≤ x / − ǫ , L ≤ K ≤ ( LK ) / . Let η k , λ ℓ , β m and ∆ B ( q ) be as in Proposition 8.3. Then we have X q ∼ Q ( q,a )=1 | ∆ B ( q ) | ≪ A,B x (log x ) A . Proof of Lemma 8.6 assuming Proposition 8.3. We consider separately the caseswhen K ∈ [ L, ( LK ) / ] and K ∈ [( LK ) / , ( LK ) / ]. If L ≤ K ≤ ( LK ) / wesee that K L + K L ≪ K L ≤ ( KL ) / ≤ x − ǫ ,K L ≤ ( KL ) / ≤ x (4 / − ǫ ) · / < x / − ǫ . Thus Lemma 8.4 gives the result in this case. If instead ( LK ) / ≤ K ≤ ( LK ) / then L ≤ ( LK ) / so QK ≤ x / − ǫ ( LK ) / < x − ǫ ,Q K L < ( x / ) ( x / ) < x ,Q K L < ( x / ) ( x / ) ( x / ) / < x . Thus Proposition 8.3 applies, giving the result in this case. This gives the result. (cid:3) Before we establish Proposition 8.1, we require some preparatory lemmas. Lemma 8.7 (Divisor function bounds) . Let | b | < x − y and y ≥ qx ǫ . Then wehave X x − y ≤ n ≤ xn ≡ a (mod q ) τ ( n ) C τ ( n − b ) C ≪ yq ( τ ( q ) log x ) O C (1) . Proof. This follows from Shiu’s Theorem [33], and is given in [4, Lemma 12]. (cid:3) Lemma 8.8 (Heath-Brown identity) . Let k ≥ and n ≤ x . Then we have Λ( n ) = k X j =1 ( − j (cid:18) kj (cid:19) X n = m ··· m j n ··· n j m , ..., m j ≤ x /k µ ( m ) · · · µ ( m j ) log n . Proof. See [26]. (cid:3) Lemma 8.9 (Small sets contribute negligibly) . Let δ > , Q ≤ x − ǫ and let A ⊆ [ x, x ] . Then we have X q ∼ Q τ ( q ) (cid:12)(cid:12)(cid:12) X n ∈A n ≡ a (mod q ) − φ ( q ) X n ∈A ( n,q )=1 (cid:12)(cid:12)(cid:12) ≪ x δ A − δ (log x ) O δ (1) . Proof. Trivially, we have that the sum is bounded by X q ∼ Q τ ( q ) (cid:16) X n ∈A n ≡ a (mod q ) φ ( q ) X n ∈A (cid:17) ≪ X n ∈A τ ( n − a ) + A X q ∼ Q τ ( q ) φ ( q ) ≪ A − δ (cid:16)X n ∼ x τ ( n − a ) /δ (cid:17) δ + A (log x ) ≪ x δ A − δ (log x ) O δ (1) + A (log x ) . Here we applied Holder’s inequality in the second line, and Lemma 8.7 in the finalline. Since A ≪ x this gives the result. (cid:3) Lemma 8.10 (Separation of variables from inequalities) . Let Q Q ≤ x − ǫ . Let N , . . . , N r ≥ z satisfy N · · · N r ≍ x . Let α n ,...,n r be a complex sequence with | α n ,...,n r | ≤ ( τ ( n ) · · · τ ( n r )) B . Then, for any choice of A > there is a constant C = C ( A, B , r ) and intervals I , . . . , I r with I j ⊆ [ P j , P j ] of length ≤ P j (log x ) − C such that X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X ∗ n ,...,n r n i ∼ N i ∀ i α n ,...,n r S n ··· n r (cid:12)(cid:12)(cid:12) ≪ r x (log x ) A + (log x ) rC X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X n ,...,n r n i ∈I i ∀ i α n ,...,n r S n ··· n r (cid:12)(cid:12)(cid:12) . Here P ∗ means that the summation is restricted to O (1) inequalities of the form n α · · · n α r r ≤ B for some constants α , . . . α r and some quantity B . The impliedconstant may depend on all such exponents α i , but none of the quantities B .Proof. Clearly we may assume that A is sufficiently large. We let implied constantsin the proof of this lemma depend on r . By Lemma 8.7, the set of n , . . . , n r with n i ∼ N i and ( τ ( n ) · · · τ ( n r )) B > (log x ) C has size ≪ ( Q i N i )(log x ) O B (1) − C ,and so by Lemma 8.9, if C is chosen sufficiently large in terms of A, B , r theseterms contribute negligibly. After rescaling (and replacing A with A + C ), we seeit suffices to consider the case when α n ,...,n k is 1-bounded.We split the summation n i ∼ N i into ⌈ (log x ) C ⌉ disjoint short intervals n i ∈ I i,j where I i,j := h N i + ( j − N i ⌈ (log x ) C ⌉ , N i + jN i ⌈ (log x ) C ⌉ (cid:17) for 1 ≤ j ≤ ⌈ (log x ) C ⌉ . We see that I i,j = [ N i,j , N i,j + N i / ⌈ (log x ) C ⌉ ) for suitable N i,j ∼ N i . We do this for each 1 ≤ i ≤ r , and so in total there are O (log Cr x )choices of these intervals.We clearly do not need to consider any tuple ( I ,j , . . . , I r,j r ) of intervals for whichthe inequalities implied by P ∗ hold for no choice of n , . . . , n r with n i ∈ I i,j i .If the inequality n α · · · n α r r ≤ B can only hold for some but not all elements( n , . . . , n r ) ∈ ( I ,j , . . . , I r,j r ) then we must have N α ,j · · · N α r r,j r = B (cid:16) O α ,...,α r (cid:16) x ) C (cid:17)(cid:17) . RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 23 If α k = 0, then, given a choice of j i for i = k , there are O α (1) choices of j k suchthat the above holds. Thus there are O α ((log x ) C ( r − ) choices of tuples of intervalsfor which the inequalities hold for some but not all elements of the intervals. ByLemma 8.9, the total contribution from these O α (log C ( r − x ) tuples of intervalswhere the inequalities sometimes hold is ≪ α x (log x ) O δ (1) − C (1 − rδ ) ≪ α x (log x ) − A if we choose δ = 1 / (2 r ) and C = C ( A, r ) large enough.Thus we only need to consider tuples ( I ,j , . . . , I r,j r ) of intervals where the inequal-ities hold for all choices of elements in the intervals. In particular, the inequalitiescan be dropped, and taking the worst tuple of intervals then gives the result. (cid:3) Lemma 8.11 (Type II estimate away from 1/2 for convolutions) . Let A > . Let x / − ǫ ≤ Q ≤ x / − ǫ , and let M , . . . , M r ≥ be such that Q ri =1 M i ≍ x and x ǫ Q < Y j ∈J M j < x / − ǫ for some set J ⊆ { , . . . , r } . Let I j ⊆ [ M j , M j ] be intervals and β (1) m , . . . , β ( r ) m be1-bounded complex sequences satisfying the Siegel-Walfisz condition (6.1) , supportedon m with P − ( m ) > z , and such that β ( j ) m = P − ( m ) >z for all m > x / .Then we have X q ∼ Q ( q,a )=1 (cid:12)(cid:12)(cid:12) X m ,...,m r m i ∈I i ∀ i (cid:16) Y ≤ j ≤ r β ( j ) m j (cid:17)(cid:16) m ··· m r ≡ a (mod q ) − ( m ··· m r ,q )=1 φ ( q ) (cid:17)(cid:12)(cid:12)(cid:12) ≪ A,r x (log x ) A . Proof of Lemma 8.11 assuming Proposition 8.3. After reordering the indices, wemay assume that J = { , . . . , k } with M ≥ · · · ≥ M k . Define N ∗ , N and M by N ∗ := x / Q / , N := k Y j =1 M j , M := r Y j = k +1 M j . Since Q < x / − ǫ we have N ∗ < x / , and since Q rj =1 M j ≍ x we have N M ≍ x .If M ≥ N ∗ then β (1) n = P − ( n ) >z , and we wish to apply Lemma 8.5 with L = M , K ≍ N/M , γ ℓ := ℓ ∈I and η k := X k = m ··· m k m i ∈I i ∀ ≤ i ≤ k β (2) m · · · β ( k ) m k , β m := X m = m k +1 ··· m r m i ∈I i ∀ k +1 ≤ i ≤ r β ( k +1) m k +1 · · · β ( r ) m r . Since M ≥ N ∗ = x / Q / and x ǫ Q < N ≪ x / − ǫ and Q ∈ [ x / − ǫ/ , x / − ǫ ],we have KL ≫ N > x ǫ Q,K Q ≪ N QLM < x / Lx / Q < x / LQ < Lx − ǫ ,KQ ≪ N Q LM < x / − ǫ LQx / < x − ǫ L, Thus Lemma 8.5 applies, and this gives the result for M ≥ N ∗ . We now consider the case when M < N ∗ . We claim we can find a product ofsome of M , . . . , M k which lies in the interval [ N / , N / ]. Since N ∗ < N / − ǫ , if M ≥ N / then either M or M · · · M k ≍ N/M will lie in this interval. If instead M < N / then all of M , . . . , M k are less than N / , so choosing j minimallysuch that M · · · M j > N / must give a suitable product (such a j exists since M · · · M k = N ). Thus in either case there is a product which lies in [ N / , N / ].After relabeling the indices { , . . . , k } (noting that this removes the property M ≥· · · ≥ M k ) we may assume that N / ≤ M · · · M J ≤ N / ≤ M J +1 · · · M k ≤ N / .We now consider L ≍ M · · · M J and K ≍ M J +1 · · · M k and α m := X m = m k +1 ··· m r m i ∈I i ∀ k +1 ≤ i ≤ r β ( k +1) m k +1 · · · β ( r ) m r , λ ℓ := X ℓ = m ··· m J m i ∈I i ∀ ≤ i ≤ J β (1) m · · · β ( J ) m J ,η k := X k = m J +1 ··· m k m i ∈I i ∀ J +1 ≤ i ≤ k β ( J +1) m J +1 · · · β ( k ) m k . We then see that since L ≤ N / ≤ K ≤ N / and LK ≍ N ≪ x / − ǫ and KL ≫ N > x ǫ Q , so Lemma 8.6 applies, giving the result for M < N ∗ . Thisfinishes the proof. (cid:3) Lemma 8.12 (Type II estimate away from x / ) . Let A > and Q Q ≤ x / − ǫ ,and let P , . . . , P J ≥ x / ǫ be such that P · · · P J ≍ x and x ǫ Q Q < Y j ∈J P j < x / − ǫ for some subset J ⊆ { , . . . , J } .Then we have X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X ∗ p ,...,p J p i ∼ P i ∀ i S p ··· p J (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A . Here P ∗ indicates that the summation is restricted by O (1) inequalities of the form p α · · · p α J J ≤ B . The implied constant may depend on all such exponents α i , butnone of the quantities B .Proof of Lemma 8.12 assuming Proposition 8.3. By Lemma 8.10, it suffices to showfor B = B ( A ) sufficiently large in terms of A X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X p ,...,p J p i ∈I i ∀ i S p ··· p J (cid:12)(cid:12)(cid:12) ≪ B x (log x ) B for all choices of intervals I , . . . , I J with I j ⊂ [ P j , P j ]. By splitting these inter-vals into sub-intervals of length P j (log x ) − B , and taking the worst such subinter-vals it suffices to show for every choice of I , . . . , I J with I j ⊆ [ P j , P j ] of length P j (log x ) − B that X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X p ,...,p J p i ∈I i ∀ i S p ··· p J (cid:12)(cid:12)(cid:12) ≪ B x (log x ) (2 J +1) B . RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 25 For p j ∈ [ N j , N j + N j (log x ) − B ) we see that Λ( p j ) = log N j + O ((log x ) − B ). Since J ≤ B is sufficiently large, Lemma 8.9 shows that the error term contributes ≪ (log x ) − B x (log x ) JB (1 − ǫ ) − O (1) ≪ x (log x ) (2 J +1) B , which is acceptable. Similarly, by Lemma 8.9, higher prime-powers contribute negli-gibly, and so we may replace the sum over primes with a sum of P − ( n ) >z Λ( n ) / log N j over integers n ∈ I j . Thus it suffices to show that X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X n ,...,n J n i ∈I i ∀ iP − ( n i ) >z ∀ i Λ( n ) · · · Λ( n J ) S n ··· n j (cid:12)(cid:12)(cid:12) ≪ B x (log x ) (2 J +1) B . We now apply Lemma 8.8 with k = 20 to expand Λ( n ) into a sum over new variables,and we put the new variables into dyadic intervals [ M i,j , M i,j ). Thus it suffices toshow that for a suitable constant C sufficiently large in terms of B X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X m , ,...,m J, m i,j ∼ M i,j ∀ i,j Q j =1 m i,j ∈I i ∀ iP − ( m i,j ) >z ∀ i,j (cid:16) Y ≤ i ≤ J ≤ j ≤ α ( j ) m i,j (cid:17) S m , ··· m J, (cid:12)(cid:12)(cid:12) ≪ C x (log x ) C . where M i,j ≤ x / for j ≤ 20 and α ( j ) m = µ ( m ) , if j ≤ , log m, if j = 21 , , if 22 ≤ j ≤ . By applying Lemma 8.10 again, we can remove the conditions Q j =1 m i,j ∈ I i .Thus we see it suffices to show that for every choice of M i,j with M i,j ≪ x / if j ≤ 20 and Q j =1 M i,j ≍ P i and for every choice of intervals I ′ i,j ⊆ [ M i,j , M i,j ] andevery C > X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X m , ,...,m J, m i,j ∈I ′ i,j ∀ i,jP − ( m i,j ) >z ∀ i,j (cid:16) Y ≤ i ≤ J ≤ j ≤ α ( j ) m i,j (cid:17) S m , ··· m J, (cid:12)(cid:12)(cid:12) ≪ C x (log x ) C . By splitting I ′ i, into shorter sub-intervals, we may replace log m i, by a constantdepending only on the subinterval in exactly the same way we saw that Λ( p i ) wasessentially constant on these short intervals. Therefore, it suffices to show for every C > X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X m , ,...,m J, m i,j ∈I ′ i,j ∀ i,j (cid:16) Y ≤ i ≤ J ≤ j ≤ ˜ α ( j ) m i,j (cid:17) S m J, ··· m ,k (cid:12)(cid:12)(cid:12) ≪ C x (log x ) C , where M i,j ≤ x / for j ≤ 20 and˜ α ( j ) m = ( P − ( m ) >z µ ( m ) , if j ≤ , P − ( m ) >z , otherwise . The result now follows from Cauchy-Schwarz (to replace the sum over q , q by asum over q ∼ Q ≍ Q Q ) and Lemma 8.11. (cid:3) Lemma 8.13 (Type II estimate near 1/2) . Let A > and let Q , Q satisfy (1.3) and (1.4) . Let P , . . . , P J ≥ x / ǫ be such that P · · · P J ≍ x and x − ǫ Q Q ≤ Y j ∈J P j ≤ x ǫ Q Q , for some subset J ⊆ { , . . . , J } .Then we have X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X ∗ p ,...,p J p i ∼ P i ∀ i S p ··· p J (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A . Here P ∗ indicates that the summation is restricted by O (1) inequalities of the form p α · · · p α J J ≤ B . The implied constant may depend on all such exponents α i , butnone of the quantities B .Proof of Lemma 8.13 assuming Proposition 8.2. This follows quickly from Propo-sition 8.2. Indeed, by Lemma 8.10, it suffices to show that X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X p ,...,p J p i ∈I i ∀ i S p ··· p J (cid:12)(cid:12)(cid:12) ≪ B x (log x ) B . for every B > I , . . . , I J with I j ⊂ [ P j , P j ]. By re-ordering the indices, we may assume J = { , . . . , k } . By considering N ≍ Q kj =1 P j ,and M ≍ Q Jj = k +1 P j and α n := X n = p ··· p k p i ∈I i , β m := X m = p k +1 ··· p J p i ∈I i , we see that Proposition 8.2 gives the result in the range Q x ǫ < N < x − ǫ Q provided (1.4) holds. This covers the range x − ǫ / ( Q Q ) < N ≤ x ǫ Q Q provided Q Q < x − ǫ . This is condition (1.3), and so gives the result. (cid:3) Putting together Lemma 8.12 and Lemma 8.12, we see Proposition 8.1 followsimmediately. Proof of Proposition 8.1 assuming Propositions 8.2 and 8.3. This follows immedi-ately from Lemma 8.12 and Lemma 8.13, noting that (1.3) and (1.4) imply that Q Q = ( Q Q ) / ( Q Q ) / < x / < x / − ǫ . (cid:3) We are left to establish Propositions 7.2, 7.4, 8.2 and 8.3. RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 27 Estimates for numbers with 4, 5 or 6 prime factors In this section we deduce Proposition 7.3 from Proposition 8.1 and Proposition 8.3.We do this by considering numbers with 4, 5 or 6 prime factors separately. Werecall that S n is defined by S n := n ≡ a (mod q q ) − ( n,q q )=1 φ ( q q ) . Proof of Proposition 7.3 assuming Proposition 8.1 and Proposition 8.3. By Lemma8.10 (and adjusting the constant A suitably), it suffices to show for every choice of I , . . . , I J with I j ⊆ [ P j , P j ] and every choice of A > X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X p ,...,p J p i ∈I i ∀ i S p ··· p J (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A . Since P i > x / ǫ and P · · · P J ≪ x we have that J ≤ 6. By assumption J ≥ J = 4 , Case 1: J = 4 . Since P i ≥ x / ǫ for all i , we have that P P P > x / ǫ , so the result followsfrom Proposition 8.1 unless P P P > x / − ǫ . Thus we may assume P < x / ǫ .Similarly, we have P P > ( P P P P ) / P / ≫ x / x / ǫ/ > x / ǫ , so weare also done by Proposition 8.1 unless P P > x / − ǫ .We now wish to apply Proposition 8.3. We let K = P , L = P , M ≍ P P , and η k := X p = kp ∈I , λ ℓ := X p = ℓp ∈I , α m := X m = p p p ∈I p ∈I . Since P ≤ · · · ≤ P ≤ x / ǫ , P P > x / − ǫ and Q Q < x / − ǫ , we have KL = P P > x / − ǫ > x ǫ Q Q ,KQ Q < x / ǫ x / − ǫ < x − ǫ . ( Q Q ) K L < x / ( P P P P ) / P / ≪ x / ǫ/ < x − ǫ , ( Q Q ) K L < x / ( P P P P ) ≪ x / < x − ǫ . Thus Proposition 8.3 applies, giving the result. Case 2: J = 5 . Since P i ≥ x / ǫ for all i , we have that P P P > x / ǫ , so the result followsfrom Proposition 8.1 unless P P P > x / − ǫ .We now wish to apply Proposition 8.3 with K ≍ P P , L = P and M ≍ P P and η k := X k = p p p ∈I p ∈I , λ ℓ := X p = ℓp ∈I , α m := X m = p p p ∈I p ∈I . We see that since P ≤ P ≤ P ≤ P ≤ P , P P P > x / − ǫ and Q Q < x / − ǫ we have Q Q x ǫ < x / − ǫ < P P P ≪ LK,Q Q K < x / ( P P P P P ) / ≪ x / < x − ǫ , ( Q Q ) K L < x / ( P P P P P ) / ≪ x / < x − ǫ , ( Q Q ) K L < x / ( P P P P P ) / ≪ x / < x − ǫ . Thus all the conditions of Proposition 8.3 are satisfied, and so Proposition 8.3 givesthe result. Case 3: J = 6 . Since all the P i are at least x / ǫ in size, we see that the product of any three ofthem is of size at least x / ǫ . But then we must have P P P ∈ [ x / ǫ , x / − ǫ ],since P P P , P P P > x / ǫ and P P P ≪ x/P P P . Therefore we see thatProposition 8.1 gives the result. (cid:3) Since we have already established Proposition 8.1 assuming Propositions 8.2 and8.3, we are left to establish Propositions 7.2, 7.4, 8.2 and 8.3. RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 29 Sieve asymptotics In this section we establish Proposition 7.2 using sieve methods, assuming the moretechnical Proposition 10.1, given below. Proposition 10.1 (Consequence of small factor type II estimate) . Let Q , Q sat-isfy (1.3) , (1.4) and (1.5) . Let α d , β e , γ m complex sequences with | α n | , | β n | , | γ n | ≤ τ ( n ) B and such that γ m satisfies the Siegel-Walfisz condition (6.1) . Assume that D, E, P satisfy x − ǫ Q / Q / < DEP, D + E + P < x / ǫ . Let M N DEP ≍ x . Then we have for every A > X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X d ∼ D X e ∼ E X p ∼ P α d β e X m ∼ M γ m ∗ X n ∼ NP − ( d ) ,P − ( n ) ≥ p S nmpde (cid:12)(cid:12)(cid:12) ≪ A,B x (log x ) A . Here P ∗ indicates that the summation is restricted by O (1) inequalities of the form p α d α e α m α n α ≤ B . The implied constant may depend on all such exponents α i , but none of the quantities B . We recall that here (and throughout the paper) S n is defined by S n := n ≡ a (mod q q ) − φ ( q q ) ( n,q q )=1 Proposition 10.1 will be established in Section 12. Lemma 10.2 (Reduction to fundamental lemma type condition) . Let z ≥ z and y ≥ . Then there are 1-bounded sequences α d , β d supported on P − ( d ) ≥ z anddepending only on d, z , z such that P − ( n ) >z = X md = nd ≤ y α d P − ( m ) >z + X n = pdmd ≤ y This follows from repeated applications of Buchstab’s identity. Let T ( n ) = P − ( n ) >z and for each r ≥ T r ( n ) := X n = p ··· p r mz z ,U r ( n ) := X n = p ··· p r mz p r ,V r ( n ) := X n = p ··· p r mz yp ··· p r − ≤ y P − ( m ) >p r . Buchstab’s identity then gives P − ( n ) >z = T ( n ) − U ( n ) − V ( n ) ,U j ( n ) = T j ( n ) − U j +1 ( n ) − V j +1 ( n ) . Repeatedly applying this and noting that U r ( n ) = T r ( n ) = V r ( n ) = 0 if r > n since 2 n > n , we find that P − ( n ) >z = X ≤ r ( − r ( T r ( n ) − V r +1 ( n )) . Thus we obtain the result by letting α d := X ≤ r ( − r X d = p ··· p r z z then X d | n λ + d = X d | n λ − d = 1 . (3) If P − ( n ) ≤ z then X d | n λ + d ≤ ≤ X d | n λ + d . (4) For any multiplicative function ω satisfying ≤ ω ( p ) ≤ κ for all primes p ,we have X d ≤ y λ ± d ω ( d ) d = Y p ≤ z (cid:16) − ω ( p ) p (cid:17) + O κ (cid:16) (log x ) − log log x (cid:17) . Proof. This is the fundamental lemma of sieve methods, and follows from [28,Lemma 6.3]. (cid:3) Lemma 10.4 (Estimates involving a smooth factor) . Let Q , Q satisfy (1.3) and (1.4) and let K, N, L satisfy KN L ≍ x and N ≤ x / ǫ x ( Q Q ) / ,x / ǫ ≤ K ≤ x / ǫ . RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 31 Let α n , η k be complex sequences with | α n | , | η n | ≤ τ ( n ) B . Then we have for every A > X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X k ∼ K X n ∼ N α n η k X ∗ ℓ ∼ LP − ( ℓ ) ≥ z S nkℓ (cid:12)(cid:12)(cid:12) ≪ A,B x (log x ) A . Here P ∗ means that the summation is restricted to O (1) inequalities of the form k α n α ℓ α ≤ B for some constants α , α , α and some quantity B , and the impliedconstant depends on all such exponents α , α , α .Proof. By Lemma 8.10, it suffices to show that for every A > X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X k ∈I K X n ∈I N X ℓ ∈I L P − ( ℓ ) ≥ z α n η k S nkℓ (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A . for all choices of intervals I K ⊆ [ K, K ], I N ⊆ [ N, N ] and I L ⊆ [ L, L ]. Since Q , Q satisfies (1.3) and (1.4), we have Q Q = ( Q Q ) / ( Q Q ) / < x / . The result follows from the Bombieri-Vinogradov Theorem if Q Q ≤ x / − ǫ , sowe may assume x / − ǫ ≤ Q Q ≤ x / . Therefore Q Q N ≤ x / ǫ ( Q Q ) / ≤ x − ǫ ,Q Q N ≤ x / ǫ ( Q Q ) / < x / − ǫ . We now wish to apply Lemma 8.5. Recalling that K < x / ǫ and L ≍ x/N K , wesee that KL ≫ xN ≫ x ǫ Q Q ,K Q Q = K Q Q Nx − ǫ x − ǫ N K ≪ x − ǫ x − ǫ Lx − ǫ < Lx − ǫ ,KQ Q = K Q Q Nx − ǫ x − ǫ N K ≪ x / / − ǫ x − ǫ Lx − ǫ < Lx − ǫ . Thus Lemma 8.5 applies, giving the result. (cid:3) Proof of Proposition 7.2 assuming Proposition 8.1 and Proposition 10.1. We may alsoassume that Q Q ≥ x / − ǫ since otherwise the result follows immediately fromthe Bombieri-Vinogradov Theorem.After expanding S p ··· p r ( x / ǫ ) as a sum of numbers with 3, 4, 5 or 6 primefactors, we see that Proposition 8.1 gives the result if there is a subproduct of P , . . . , P r which lies in [ x / ǫ , x / − ǫ ]. In particular, since P · · · P r ≤ x / and P r ≥ x / ǫ , this gives the result unless P · · · P r − ≤ x / ǫ , which we nowassume.By Lemma 8.10, it suffices to show that for every choice of I i ⊆ [ P i , P i ] and everychoice of B > X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X p ,...,p r p i ∈I i S p ··· p r ( x / ǫ ) (cid:12)(cid:12)(cid:12) ≪ B x (log x ) B . We recall that z := x / ǫ and S p ··· p r ( z ) = X m ∼ x/ ( p ··· p r ) P − ( m ) >z S mp ··· p r . Let y := x − ǫ / ( Q Q ) / . Since Q Q ≥ x / − ǫ , we see that y ≤ z . By Lemma10.2 (taking y = y , z = z ), it suffices to show that for every choice of 1-boundedcoefficients α d , β d we have X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X p ,...,p r p i ∈I i X z y β d X n ∼ x/ ( dp ′ p ··· p r ) P − ( n ) ≥ p S ndp ′ p ··· p r (cid:12)(cid:12)(cid:12) ≪ x (log x ) B , X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X p ,...,p r p i ∈I i X d ≤ y α d X m ∼ x/ ( dp ··· p r ) P − ( m ) ≥ z S mdp ··· p r (cid:12)(cid:12)(cid:12) ≪ x (log x ) B . We begin by considering the first expression above. We note that dp ′ > y and d, p ′ < z . Thus, by letting e = 1 and m = p · · · p r , we see that Proposition 10.1gives the first estimate.Similarly, by letting k = p r , ℓ = m and n = dp · · · p r − , we see that Lemma 10.4gives the second result if P r ≤ x / . If instead P r ≥ x / then we must have that r = 1.Finally, if r = 1 then we note that since Q Q ≥ x / − ǫ , for any d , p occurring wemust have dp ≤ x / ǫ x − ǫ ( Q Q ) / ≪ x − ǫ Q Q . Thus the range of summation for m has length at least x ǫ Q Q . But then byLemma 10.3, we have that for any choice of B > X m ∼ x/ ( dp ) P − ( m ) ≥ z S mdp ≤ X m ∼ x/ ( dp ) m ≡ adp (mod q q ) X e | me ≤ y λ + d − φ ( q q ) X m ∼ x/ ( dp )( m,q q )=1 X e | me ≤ y λ − e ≤ X e ≤ y ( e,q q )=1 λ + e (cid:16) xq q dp e + O (1) (cid:17) − φ ( q q ) X e ≤ y ( e,q q )=1 λ − e (cid:16) φ ( q q ) xq q dp e + O ( τ ( q q )) (cid:17) ≪ B xq q p d (log x ) B , and similarly we obtain a lower bound by swapping the roles of λ + e and λ − e . Thusthe inner sum over m is suitably small that these terms contribute negligibly when P r ≥ x / . This gives the result. (cid:3) Since we have already established Proposition 8.1 assuming Propositions 8.2 and8.3, we are left to establish Propositions 7.4, 8.2, 8.3 and 10.1. RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 33 Estimates for numbers with 3 prime factors In this section we reduce the proof of Proposition 7.4 to Proposition 11.1, whereessentially the indicator function of the primes is replaced by a smooth weight.We do this using using Proposition 10.1 and repeated Buchstab iterations. Thefollowing is our key proposition, which will be established later in Section 20. Proposition 11.1 (Estimate for triple divisor function) . Let A > . Let Q, R satisfy Q R < x , Q R < x / , QR < x / . Let x / ≥ N ≥ N ≥ N ≥ x ǫ and M ≥ x ǫ satisfy N N N M ≍ x and M < x − ǫ ( QR ) / . Let | α m | ≤ τ ( m ) B be a complex sequence, let I , I , I be intervals with I j ⊆ [ N j , N j ] , and let ∆ K ( q ) := X m ∼ M α m X n ∈I n ∈I n ∈I P − ( n n n ) ≥ z (cid:16) mn n n ≡ a (mod q ) − ( mn n n ,q )=1 φ ( q ) (cid:17) . Then we have X q ∼ Q ( q,a )=1 X r ∼ R ( r,a )=1 | ∆ K ( qr ) | ≪ A x (log x ) A . Proof of Proposition 7.4 assuming Propositions 11.1, 7.3 and 10.1. We recall that S n is defined by S n := n ≡ a (mod q q ) − φ ( q q ) ( n,q q )=1 By Lemma 8.10, it suffices to show that for every choice of I i ⊆ [ P i , P i ] and every A > X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X p ,p ,p p i ∈I i ∀ i S p p p (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A . We recall z = x / ǫ and P i ≤ x / ǫ , so any m ∈ I i can have at most 2 primefactors bigger than z . Thus, by three applications of Buchstab’s identity X p ,p ,p p i ∈I i ∀ i S p p p = X p ∈I p ∈I m ∈I P − ( m ) >z S p p m − X p ∈I p ∈I X z z ∀ i S m m m − X p ∈I p ∈I X z z X z z X z z ∀ i S m m m (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A . Our aim is to use Lemma 10.2 and Proposition 10.1 to show that it is sufficient toreplace the condition P − ( m i ) > z with the condition P − ( m i ) > z . If we can dothis, then the result follows from Proposition 11.1.We set y := x − ǫ / ( Q Q ) / ≤ z . By Lemma 10.2 (with y = y and z = z )applied to m , it suffices to show that for every 1-bounded sequence α d and β d wehave(11.1) X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:16) | V (1) ( q q ) | + | T (1) ( q q ) | (cid:17) ≪ B x (log x ) B , where V (1) ( q q ) := X m ∈I m ∈I P − ( m m ) >z X z p d p >y β d X d p n ∈I P − ( n ) ≥ p S n d p m m ,T (1) ( q q ) := X m ∈I m ∈I P − ( m m ) >z X d ≤ y α d X d n ∈I P − ( n ) ≥ z S n d m m . We let e = 1, d = d p = p , ℓ = n and m = m m . Since we only consider d p > y , and d , p ≤ z , we see that Proposition 10.1 implies that the V (1) ( q q )terms contribute negligibly to (11.1). Thus we are left to consider the contributionfrom the T (1) ( q q ) terms. RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 35 We now apply Lemma 10.2 to the m sum in T (1) ( q q ), this time taking y = y /d .Thus it suffices to show for every 1-bounded sequence α ′ d and β ′ d that(11.2) X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:16) | V (2) ( q q ) | + | T (2) ( q q ) | (cid:17) ≪ B x (log x ) B , where V (2) ( q q ) := X d ,n ,n ,m m ∈I d ≤ y n d ∈I P − ( n ) >z P − ( m ) >z α d X z p d d p >y β ′ d X d p n ∈I P − ( n ) ≥ p S n d n d p m ,T (2) ( q q ) := X d ≤ y X n d ∈I P − ( n ) >z X m ∈I P − ( m ) >z X d ≤ y /d α d α ′ d X d n ∈I P − ( n ) ≥ z S n d n d m . Again, on letting e = d , p = p , d = d , n = n and m = n m , we see thatProposition 10.1 shows that the V (2) ( q q ) contribute suitably little to (11.2), sowe are left to consider the T (2) terms.Finally, we apply Lemma 10.2 once more to the m summation of T (2) ( q q ) with y = y /d d . Thus it suffices to show for every 1-bounded sequence α ′′ d and β ′′ d that(11.3) X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:16) | V (3) ( q q ) | + | T (3) ( q q ) | (cid:17) ≪ B x (log x ) B , where V (3) ( q q ) := X d ,d ,n ,n d d ≤ y n d ∈I n d ∈I P − ( n n ) >z α d α ′ d X z p d d d p >y β ′′ d X d p n ∈I P − ( n ) ≥ p S n n n d d d p ,T (3) ( q q ) := X d d d ≤ y X n d ∈I n d ∈I P − ( n n ) >z α d α ′ d α ′′ d X d n ∈I P − ( n ) ≥ z S n n n d d d . Again, on letting e = d d , p = p , d = d , n = n and m = n n , we see thatProposition 10.1 shows that the V (3) terms contribute suitably little to (11.3), sowe are left to consider the T (3) terms.Finally, we split the T (3) ( q q ) sum by putting d , d , d into short intervals d i ∈J i = [ D i , D i (1 + log − C x )) for a suitably large constant C = C ( A ). We thensee by Lemma 8.9 we may replace the constraint d i n i ∈ I i by D i n i ∈ I i since thecontribution from near the end points is negligible. Therefore it suffices to showthat for every B > X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X d ,d ,d d d d ≤ y d i ∈J i ∀ i α d α ′ d α ′′ d X n ,n ,n D i n i ∈I i ∀ iP − ( n i ) >z ∀ i S d d d n n n (cid:12)(cid:12)(cid:12) ≪ B x (log x ) B . Thus, on letting α m := X m = d d d d d d ≤ y d i ∈J i ∀ i α d α ′ d α ′′ d , and dividing the m -summation into dyadic intervals, we see that (11.4) follows fromProposition 11.1, as required. This gives the result. (cid:3) We are left to establish Propositions 8.2, 8.3, 10.1 and 11.1. RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 37 Small factor Type II estimate In this section we reduce Proposition 10.1 to two technical propositions aboutconvolutions, namely Proposition 12.1 and Proposition 12.2, given below.The first key proposition is a result which is a generalization of work of Fouvry [14].This will be established in Section 15. Proposition 12.1 (Fouvry-style estimate) . Let A > and C = C ( A ) be sufficientlylarge in terms of A . Assume that N, M, Q , Q satisfy N M ≍ x and x ǫ Q < N,N Q Q < x − ǫ ,N Q Q < x − ǫ ,N Q Q < x / − ǫ ,Q Q < x / − ǫ N / . Let β m , α n be complex sequences such that | α n | , | β n | ≤ τ ( n ) B and such that α n satisfies the Siegel-Walfisz condition (6.1) and α n is supported on n with all primefactors bigger than z := x / (log log x ) . Let ∆( q ) := X m ∼ M X n ∼ N α n β m (cid:16) mn ≡ a (mod q ) − ( mn,q )=1 φ ( q ) (cid:17) . Then we have X q ∼ Q X q ∼ Q ( q q ,a )=1 | ∆( q q ) | ≪ A,B x (log x ) A . Our second estimate is a new convolution result tailored for when one factor is afairly small power of x . This will be established in Section 16. Proposition 12.2 (Small divisor estimate) . Let A > and C = C ( A ) be suffi-ciently large in terms of A . Assume that N, M, Q , Q satisfy N M ≍ x and N Q Q < x − ǫ ,Q Q < x − ǫ N. Let β m , α n be complex sequences such that | α n | , | β n | ≤ τ ( n ) B and such that α n satisfies the Siegel-Walfisz condition (6.1) and α n is supported on n with all primefactors bigger than z := x / (log log x ) . Let ∆( q ) := X m ∼ M X n ∼ N α n β m (cid:16) mn ≡ a (mod q ) − ( mn,q )=1 φ ( q ) (cid:17) . Then we have X q ∼ Q X q ∼ Q ( q q ,a )=1 | ∆( q q ) | ≪ A,B x (log x ) A . Putting together Proposition 12.1 and Proposition 12.2, we obtain the following. Lemma 12.3 (Small Factor Type II estimate for convolutions) . Let Q , Q satisfy Q Q < x − ǫ ,Q Q < x / − ǫ ,Q Q < x − ǫ ,Q Q < x / − ǫ . Let N , M be such that N M ≍ x and x − ǫ ( Q Q ) / < N < x / ǫ . Let β m , α n be complex sequences such that | α n | , | β n | ≤ τ ( n ) B and such that α n satisfies the Siegel-Walfisz condition (6.1) and α n is supported on n with all primefactors bigger than z := x / (log log x ) . Let ∆( q ) := X m ∼ M X n ∼ N α n β m (cid:16) mn ≡ a (mod q ) − ( mn,q )=1 φ ( q ) (cid:17) . Then we have X q ∼ Q X q ∼ Q ( q q ,a )=1 | ∆( q q ) | ≪ A,B x (log x ) A . Proof of Lemma 12.3 assuming Proposition 12.1 and Proposition 12.2. We only needconsider Q Q ≥ x / − ǫ because otherwise the result is immediate from the Bombieri-Vinogradov Theorem. Proposition 12.1 gives the result providedmax (cid:16) Q Q x − ǫ , Q x ǫ (cid:17) < N < min (cid:16) x / − ǫ Q Q , x / − ǫ Q / Q / , x / − ǫ Q Q / (cid:17) . We can also apply Proposition 12.1 with a trivial factorization by taking ‘ Q ’ to be Q Q and ‘ Q ’ to be 1. This simplifies to gives the result in the range Q Q x − ǫ < N < x / − ǫ Q / Q / . (Here we used the fact that x / − ǫ / ( Q Q ) / < x / − ǫ / ( Q Q ) / , x / − ǫ / ( Q Q )since Q Q > x / − ǫ .) Finally, we can also apply Proposition 12.2 (with Q and Q swapped). This gives the result in the range Q Q x − ǫ < N < x / − ǫ Q / Q / . We see that if(12.1) Q / Q / < x / − ǫ , then Q Q x − ǫ < x / − ǫ Q / Q / , and so the final two ranges overlap. Similarly, if(12.2) Q / Q / < x / − ǫ , RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 39 then Q x ǫ < x / − ǫ Q / Q / , and so the first two ranges overlap. Thus if (12.1) and (12.2) both hold we havethe result for Q Q x − ǫ < N < min (cid:16) x / − ǫ Q Q , x / − ǫ Q / Q / , x / − ǫ Q Q / (cid:17) . This gives the result in the entire range provided we have Q Q x − ǫ < x − ǫ ( Q Q ) / , x / ǫ < x / − ǫ Q Q ,x / ǫ < x / − ǫ Q / Q / , x / ǫ < x / − ǫ Q Q / . These conditions hold if we have Q Q < x − ǫ , (12.3) Q Q < x / − ǫ , (12.4) Q Q < x / − ǫ , (12.5) Q Q < x / − ǫ . (12.6)Finally, we note that if Q Q ≥ x / − ǫ and (12.5) holds, then Q Q = ( Q Q ) ( Q Q ) < x / − ǫ x − ǫ < x / − ǫ , and so (12.2) automatically holds. Similarly, Q Q = ( Q Q ) / ( Q Q ) / < x (96+8 / / < x / − ǫ , so if (12.3) and (12.5) both hold then (12.6) automatically holds. This gives theresult. (cid:3) Proof of Proposition 10.1 assuming Proposition 12.1 and Proposition 12.2. We firstnote that our assumptions on Q , Q imply that Q Q = ( Q Q ) / ( Q Q ) / < x / < x − ǫ ,Q Q = ( Q Q ) / ( Q Q ) / < x / − ǫ ,Q Q = ( Q Q ) / ( Q Q ) / < x / < x / − ǫ . Thus Q , Q satisfy all the conditions of Lemma 12.3. By Lemma 8.10, we see thatit is sufficient to show that X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X d ∈I X e ∈I X p ∈I α d β e X m ∈I γ m X n ∈I P − ( d ) ,P − ( n ) ≥ p S nmpde (cid:12)(cid:12)(cid:12) ≪ B x (log x ) B . for every choice of B > I , . . . , I with I ⊆ [ D, D ], I ⊆ [ E, E ], I ⊆ [ P, P ], I ⊆ [ M, M ] and I ⊆ [ N, N ]. We subdivide I into O (log C x ) subintervals of length P/ log C x , for a suitably large constant C = C ( B ).Taking the worst such interval, we see it suffices to show that X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X d ∈I X e ∈I X p ∈I α d β e X m ∈I γ m X n ∈I P − ( d ) ,P − ( n ) ≥ p S nmpde (cid:12)(cid:12)(cid:12) ≪ C x (log x ) C/ , for every interval I = [ P ′ , P ′ + P/ log C x ] with P ′ ∈ [ P, P ]. We now replacethe conditions P − ( d ) , P − ( n ) ≥ p with P − ( d ) , P − ( n ) ≥ P ′ . This introduces anerror from the contribution of when P − ( d ) ∈ I or P − ( m ) ∈ I . Since there are O ( DN/ log C x ) such choices of d, n and there are O ( P/ log C x ) choices of p , we seethat this counts contributions S r from r coming from a set of size ≪ x/ log C x .By Lemma 8.9, this contribution is acceptably small. Thus it suffices to show X q ∼ Q X q ∼ Q ( q q ,a )=1 (cid:12)(cid:12)(cid:12) X d ∈I P − ( d ) ≥ P ′ X e ∈I X p ∈I α d β e X m ∈I γ m X n ∈I P − ( n ) ≥ P ′ S nmpde (cid:12)(cid:12)(cid:12) ≪ C x (log x ) C/ , Since we have D, E, P ≤ x / ǫ and DEP > x − ǫ / ( Q Q ) / there is somesub–product of D, E, P which lies in [ x − ǫ / ( Q Q ) / , x / ǫ ]. Grouping thecorresponding variables together, we then see that the result now follows fromLemma 12.3. (cid:3) We have already established Proposition 7.3 assuming Proposition 8.1 and 8.3. Wehave already established Proposition 8.1 assuming Proposition 8.2 and 8.3. We havealready established Proposition 10.1 assuming Propositions 12.1 and 12.2. Thus weare left to establish Propositions 8.2, 8.3, 11.1, 12.1 and 12.2. RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 41 Preparatory lemmas The remainder of the paper is now dedicated to establishing the more technicalPropositions 8.2, 8.3, 11.1, 12.1 and 12.2 individually. Before we embark on this,we establish various lemmas which we will make use of in the later sections. Lemma 13.1 (Bezout’s identity) . Let ( q , q ) = 1 . Then we have aq q = aq q + aq q (mod 1) . Proof. Multiplying through by q q , we see that this follows from the fact that q q + q q ≡ q q ), which is obvious from the Chinese remainder theoremand considering the equation (mod q ) and (mod q ) separately. (cid:3) Lemma 13.2 (Splitting into coprime sets) . Let N ⊆ Z > be a set of pairs ( a, b ) satisfying:(1) a, b ≤ x O (1) ,(2) gcd( a, b ) = 1 ,(3) The number of prime factors of a and of b is ≪ (log log x ) .Then there is a partition N = N ⊔ N ⊔ · · · ⊔ N J into J disjoint subsets with J ≪ exp (cid:16) O (log log x ) (cid:17) , such that if ( a, b ) and ( a ′ , b ′ ) are in the same set N j , then gcd( a, b ′ ) = gcd( a ′ , b ) = 1 .Proof. This follows immediately from [14, Lemme 6]. (cid:3) Lemma 13.3 (Weil bound for Kloosterman sums) . Let S ( m, n ; c ) be the standardKloosterman sum S ( m, n ; c ) := X b (mod c )( b,c )=1 e (cid:16) mb + nbc (cid:17) . Then we have that S ( m, n ; c ) ≪ τ ( c ) c / gcd( m, n, c ) / . Proof. This is [28, Corollary 11.12]. (cid:3) Lemma 13.4 (Poisson Summation) . Let C > and f : R → R be a smoothfunction which is supported on [ − , and satisfies k f ( j ) k ∞ ≪ j (log x ) jC for all j ≥ , and let M, q ≤ x . Then we have X m ≡ a (mod q ) f (cid:16) mM (cid:17) = Mq b f (0) + Mq X ≤| h |≤ H b f (cid:16) hMq (cid:17) e (cid:16) ahq (cid:17) + O C ( x − ) , for any choice of H > qx ǫ /M . Moreover, if f satisfies k f ( j ) k ∞ ≪ (( j + 1) log x ) jC for all j ≥ then we have the same result with H ≥ q (log x ) C +1 /M . In particular, X m ≡ a (mod q ) ψ (cid:16) mM (cid:17) = Mq b ψ (0) + Mq X ≤| h |≤ H b ψ (cid:16) hMq (cid:17) e (cid:16) ahq (cid:17) + O ( x − ) , for any choice of H ≥ q (log x ) /M .Proof. This is a truncated Poisson summation formula. Let g ( n ) = f (( a + qn ) /M ).Then by Poisson summation we have X m ≡ a (mod q ) f (cid:16) mM (cid:17) = X n ∈ Z g ( n ) = X h ∈ Z b g ( h ) = Mq X h ∈ Z b f (cid:16) hMq (cid:17) e (cid:16) haq (cid:17) . If k f ( j ) k ∞ ≪ j (log x ) jC then by integration by parts we see that | b f ( t ) | ≪ j (log x ) jC t − j for any j ≥ 1. Choosing j large enough we see that the terms with | h | ≥ qx ǫ /M contribute O C ( x − ), which gives the first claim. If k f ( j ) k ∞ ≪ (( j + 1) log x ) jC we see the terms with | h | > q (log x ) C +1 /M contribute ≪ C Mq X | h | >q (log x ) C +1 /M ( j log x ) jC | πhM/q | j ≪ C ( j log x ) jC | π log C +1 x | j − . Choosing j = ⌈ log x ⌉ then gives the result. (cid:3) Lemma 13.5 (Completion of inverses) . Let C > and f : R → R be a smoothfunction which is supported on [ − , and satisfies k f ( j ) k ∞ ≪ j (log x ) jC for all j ≥ . Let ( d, q ) = 1 . Then we have for any H ≥ x ǫ dq/N X ( n,q )=1 n ≡ n (mod d ) f (cid:16) nN (cid:17) e (cid:16) bnq (cid:17) = N b f (0) dq X ( c,q )=1 e (cid:16) bcq (cid:17) + Ndq X ≤| h |≤ H b f (cid:16) hNdq (cid:17) e (cid:16) n qhd (cid:17) X c (mod q )( c,q )=1 e (cid:16) bdc + hcq (cid:17) + O C ( x − ) . Moreover, if k f ( j ) k ∞ ≪ (( j + 1) log x ) jC then we have the same result for any H ≥ (log x ) C +1 dq/N .Proof. We first put n into residue classes (mod dq ) and the apply Lemma 13.4.This gives X ( n,q )=1 n ≡ n (mod d ) f (cid:16) nN (cid:17) e (cid:16) bnq (cid:17) = X c (mod q )( c,q )=1 e (cid:16) bcq (cid:17) X n ≡ c (mod q ) n ≡ n (mod d ) f (cid:16) nN (cid:17) = N b f (0) dq X c (mod q )( c,q )=1 e (cid:16) bcq (cid:17) + Ndq X c (mod q )( c,q )=1 e (cid:16) bcq (cid:17) X ≤| h |≤ H b f (cid:16) hNdq (cid:17) e (cid:16) h (cid:16) cdq + n qd (cid:17)(cid:17) + O C ( x − ) . Here we used the Chinese Remainder Theorem to combine the congruence for n inthe final line. A change of variables in the c -summations then gives the result. (cid:3) Lemma 13.6 (Summation with coprimality constraint) . Let C > and f : R → R be a smooth function which is supported on [ − , and satisfies k f ( j ) k ∞ ≪ j RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 43 (log x ) jC for all j ≥ . Then we have X ( m,q )=1 f (cid:16) mM (cid:17) = φ ( q ) q M + O ( τ ( q )(log x ) C ) . Proof. We use M¨obius inversion to rewrite the condition ( m, q ) = 1. X ( m,q )=1 f (cid:16) mM (cid:17) = X d | q µ ( d ) X n f (cid:16) dnM (cid:17) = X d | q µ ( d ) (cid:16) M b f (0) d + O (log C x ) (cid:17) = φ ( q ) q M b f (0) + O ( τ ( q ) log C x ) . Here we used Poisson summation and the fact b f ( t ) ≪ (log x ) C /t from k f (2) k ∞ ≪ (log x ) C in the final line. (cid:3) Lemma 13.7. Let C, B > be constants and let α n be a sequence satisfing theSiegel-Walfisz condition (6.1) , supported on n ≤ x with P − ( n ) ≥ z = x / (log log x ) and satisfying | α n | ≤ τ ( n ) B . Then τ ( n ) ≤ (log x ) C α n also satisfies the Siegel-Walfiszcondition.Proof. since α n satisfies the Siegel-Walfisz condition, we just need to show that τ ( n ) ≥ (log x ) C α n also does. We see that for any choice of A > X n ∼ Nn ≡ a (mod q ) τ ( n ) ≥ (log x ) C α n ≤ X n ≤ min(2 x,N ) P − ( n ) ≥ z τ ( n ) B + A (log x ) AC ≤ N (log x ) AC Y z andlet α n be a complex sequence which satisfies the Siegel-Walfisz condition (6.1) andsatisfies | α n | ≤ τ ( n ) B . Then for any A > there is a constant C = C ( A, B ) suchthat if Q < N/ (log N ) C we have X q ≤ Q τ ( q ) B X b (mod q )( b,q )=1 (cid:12)(cid:12)(cid:12) X n ∼ N α n (cid:16) n ≡ b (mod q ) − ( n,q )=1 φ ( q ) (cid:17)(cid:12)(cid:12)(cid:12) ≪ A,B N (log N ) A . Proof. Without the factor τ ( q ) this is part (a) of [3, Theorem 0]. By Cauchy-Schwarz we have X q ≤ Q τ ( q ) B X b (mod q )( b,q )=1 (cid:12)(cid:12)(cid:12) X n ∼ N α n (cid:16) n ≡ b (mod q ) − ( n,q )=1 φ ( q ) (cid:17)(cid:12)(cid:12)(cid:12) ≪ S / S / , where, bounding trivially via Lemma 8.7 S := X q ≤ Q τ ( q ) B X b (mod q )( b,q )=1 (cid:12)(cid:12)(cid:12) X n ∼ N α n (cid:16) n ≡ b (mod q ) − ( n,q )=1 φ ( q ) (cid:17)(cid:12)(cid:12)(cid:12) ≪ N X q ≤ Q ( τ ( q ) log x ) O B (1) q ≪ N (log x ) O B (1) , and by [3, Theorem 0] for any constant C > S := X q ≤ Q X b (mod q )( b,q )=1 (cid:12)(cid:12)(cid:12) X n ∼ N α n (cid:16) n ≡ b (mod q ) − φ ( q ) (cid:17)(cid:12)(cid:12)(cid:12) ≪ C N (log N ) C . Choosing C sufficiently large in terms of A and B gives the result. (cid:3) Lemma 13.9 (Most moduli have small square-full part) . Let Q < x − ǫ . Let γ b , c q be complex sequences satisfying | γ b | , | c b | ≤ τ ( b ) B . Let sq ( n ) denote the square-fullpart of n . (i.e. sq ( n ) = Q p : p | n p ν p ( n ) ). Then for every A > there is a constant C = C ( A, B ) such that X q ∼ Qsq ( q ) ≥ (log x ) C c q X b ≤ x γ b (cid:16) b ≡ a (mod q ) − ( b,q )=1 φ ( q ) (cid:17) ≪ A,B x (log x ) A . Proof. By Lemma 8.7 we have that X b ≤ x γ b (cid:16) b ≡ a (mod q ) − ( b,q )=1 φ ( q ) (cid:17) ≪ x (log x ) O B (1) q + X b ≤ xb ≡ a (mod q ) τ ( b ) B ≪ x ( τ ( q ) log x ) O B (1) q . Let q = q q where sq ( q ) = q , and let q be the radical of q , so that q | q . Then,using Lemma 8.7 again, we see that the sum in the lemma is bounded by X q ≥ (log x ) C p | q ⇒ p | q X q ≤ Q/q x ( τ ( q q ) log x ) O B (1) q q ≪ x (log x ) O B (1) X q ≥ (log x ) C/ τ ( q ) O B (1) q ≪ x (log x ) O B (1) (log x ) C/ . Choosing C large enough then gives the result. (cid:3) Lemma 13.10 (Most moduli have small z -smooth part) . Let Q < x − ǫ . Let γ b , c q be complex sequences with | γ b | , | c b | ≤ τ ( n ) B and recall z := x / (log log x ) and y := x / log log x . Let sm( n ; z ) denote the z -smooth part of n . (i.e. sm( n ; z ) = Q p ≤ z p ν p ( n ) ). Then for every A > we have that X q ∼ Q sm( q ; z ) ≥ y c q X b ≤ x γ b (cid:16) b ≡ a (mod q ) − ( b,q )=1 φ ( q ) (cid:17) ≪ A,B x (log x ) A . RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 45 Proof. As in the proof of Lemma 13.9, by Lemma 8.7 we have the trivial bound X b ≤ x γ b (cid:16) b ≡ a (mod q ) − ( b,q )=1 φ ( q ) (cid:17) ≪ x ( τ ( q ) log x ) O B (1) q . We factor q = q q into z -smooth and z -rough parts. Thus we see that the sumin the lemma is bounded by X q ≥ y P + ( q ) ≤ z X q ∼ Q x ( τ ( q q ) log x ) O B (1) q q ≪ x (log x ) O B (1) X q ≥ y P + ( q ) ≤ z τ ( q ) O B (1) q . Now we us Rankin’s trick. Letting η = 1 / log z = 1 / (log log x ) , we have X q ≥ y P + ( q ) ≤ z τ ( q ) B q ≤ y − η X P + ( q ) ≤ z τ ( q ) B q − η ≪ B y − η Y p ≤ z B ≤ p (cid:16) − B p − η (cid:17) − ≪ (log z ) O B (1) y − / log z . Since y − / log z ≪ A (log z ) − A for every A > 0, this gives the result. (cid:3) Dispersion estimates In this section we perform the initial steps of the Linnik dispersion method (see[30]) to reduce the main problem to that of estimating certain exponential sums.The argument is very similar to that of [3, § § Lemma 14.1 (First dispersion sum) . Let η q,d,r and α n be bounded complex se-quences such that | η q,d,r | ≤ τ ( qdr ) B and | α n | ≤ τ ( n ) B . Let S := X e ∼ E µ ( e ) X q ( q,a )=1 X d ∼ D ( d,a )=1 X r ,r ∼ R ( r r ,ae )=1 ψ (cid:16) qQ (cid:17) φ ( qde ) η q,d,r η q,d,r φ ( qd ) φ ( qder ) φ ( qder ) × X n ,n ∼ N ( n ,qder )=1( n ,qder )=1 α n α n X m ( m,qdr r )=1 ψ (cid:16) mM (cid:17) , and for C = C ( A, B ) sufficiently large in terms of A and B , let M > (log x ) C . Then S = S MT + O A,B (cid:16) M N QD (log x ) A (cid:17) , where S MT := M b ψ (0) X e ∼ E µ ( e ) X q ( q,a )=1 ψ (cid:16) qQ (cid:17) X d ∼ D ( d,a )=1 X r ,r ∼ R ( r r ,ae )=1 × φ ( qde ) η q,d,r η q,d,r φ ( qdr r ) φ ( qd ) φ ( qder ) φ ( qder ) qdr r X n ,n ∼ N ( n ,qder )=1( n ,qder )=1 α n α n . Proof. We consider the inner sum of S . By Lemma 13.6 we have that X m ( m,qdr r )=1 ψ (cid:16) mM (cid:17) = φ ( qdr r ) qdr r M b ψ (0) + O ( τ ( qdr r )) . By Lemma 8.7, the error term above contributes to S a total ≪ X e ∼ E X q ≍ Q X d ∼ D X r ,r ∼ R X n ,n ∼ N τ ( qdr r ) B +1 τ ( n ) B τ ( n ) B log xQ D R E ≪ N (log x ) O B (1) QD , which is O A ( M N / ( QD (log x ) A )) provided(14.1) M > (log x ) C and C is sufficiently large in terms of A and B . The main term above contributes S MT to S , and this gives the result. (cid:3) RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 47 Lemma 14.2 (Second dispersion sum) . Let η q,d,r and α n be complex sequencessuch that | η q,d,r | ≤ τ ( qdr ) B and | α n | ≤ τ ( n ) B and such that α n is supported on P − ( n ) ≥ z and η q,d,r is supported on P − ( r ) ≥ z . Let S := X e ∼ E µ ( e ) X q ( q,a )=1 X d ∼ D ( d,a )=1 X r ,r ∼ R ( r r ,ae )=1 ψ (cid:16) qQ (cid:17) η q,d,r η q,d,r φ ( qder ) × X n ,n ∼ N ( n ,qder )=1( n ,qder )=1 α n α n X mmn ≡ a (mod qdr )( m,r )=1 ψ (cid:16) mM (cid:17) . Then we have S = S MT + O ( M | E | ) + O A,B (cid:16) M N QD (log x ) A (cid:17) , where S MT is as given by Lemma 14.1, H := QDRM log x and E is given by E := X e ∼ E µ ( e ) X q ( q,a )=1 X d ∼ D ( d,a )=1 X r ,r ∼ R ( r r ,a )=1 ψ (cid:16) qQ (cid:17) η q,d,r η q,d,r φ ( qder ) qdr × X n ,n ∼ N ( n ,qder )=1( n ,qder )=1 α n α n X ≤| h |≤ H b ψ (cid:16) hMqdr (cid:17) e (cid:16) ahn qdr (cid:17) . Proof. For notational simplicity, we let ℓ = qd throughout the proof. First weremove the condition ( m, r ) = 1 from S , which introduces additional terms whichcontribute a total ≪ X f> X e ∼ E X n ,n ∼ N X m ≪ Mf | m X ℓ | mn − a X r | mn − a X r ∼ Rf | r P − ( r ) ≥ z τ ( n ) B τ ( n ) B τ ( ℓr r ) B log xQDER ≪ X f>z N (log x ) O B (1) QDf X n ∼ N X m ≪ Mf | m τ ( mn − a ) B ≪ M N (log x ) O B (1) QDz . Here we used the fact that η q,d,r is supported on P − ( r ) ≥ z in the first line, andso this implies that any divisor f > r is bigger than z in the second line, andwe used Lemma 8.7 in the final line. This contribution is negligible compared with M N / ( QD log A x ), so it suffices to consider S without the constraint ( m, r ) = 1. Similarly, we remove the constraint ( e, r r ) = 1, which introduces additional termswhich contribute X f f >z X e ∼ Ef f | e X n ,n ∼ N X m ≪ Mmn ≡ a (mod f ) X ℓ | mn − a X r | mn − af | r X r ∼ R f | r τ ( n ) B τ ( n ) B τ ( ℓr r ) B log xQDER ≪ X f f >z Ef f N (cid:16) Mf + 1 (cid:17) (log x ) O B (1) Rf QDER ≪ M N (log x ) O B QDz + N (log x ) O B QD , which is acceptably small. Thus we can drop this constraint too.By Lemma 13.4, we have X mmn ≡ a (mod ℓr ) ψ (cid:16) mM (cid:17) = Mℓr b ψ (0) + Mℓr X ≤| h |≤ H b ψ (cid:16) hMℓr (cid:17) e (cid:16) ahn ℓr (cid:17) + O ( x − ) , where H := QDRM log x. The final term above clearly contributes a negligible amount. The first term abovecontributes to S a total M b ψ (0) X e ∼ E µ ( e ) X q ( q,a )=1 X d ∼ D ( d,a )=1 X r ,r ∼ R ( r r ,a )=1 ψ (cid:16) qQ (cid:17) η q,d,r η q,d,r φ ( qder ) qdr X n ,n ∼ N ( n ,qder )=1( n ,qder )=1 α n α n . This differs from S MT by a factor φ ( qde ) φ ( qdr r ) φ ( qd ) φ ( qder ) r in the summand, and misses the condition ( e, r r ) = 1. Again, since η q,d,r issupported on P − ( r ) ≥ z , and r ≪ x , we see that the factor above is Y p | r r (cid:16) O (1) p (cid:17) = (cid:16) O (cid:16) z (cid:17)(cid:17) O (log x ) = 1 + O (cid:16) log xz (cid:17) . The error term O ((log x ) /z ) contributes a total ≪ M log xz X e ∼ E X r ,r ∼ R X ℓ ≪ QD log xQ D R E X n ,n ∼ N τ ( n n ℓr r ) B ≪ M N (log x ) O B (1) QDz . Finally, we may reintroduce the condition ( e, r r ) = 1 in exactly the same waythat we originally removed it, at the cost of a negligible error. Thus, we see that(14.2) S = S MT + O A,B (cid:16) M N QD (log x ) A (cid:17) + O ( M | E | ) , where E is the middle term contribution, given in the statement of the lemma. (cid:3) Lemma 14.3 (Third dispersion sum) . Let η q,d,r and α n be complex sequences suchthat | η q,d,r | ≤ τ ( qdr ) B and | α n | ≤ τ ( n ) B , η q,d,r is supported on P − ( r ) ≥ z , α n RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 49 is supported on P − ( n ) ≥ z and such that α n satisfies the Siegel-Walfisz condition (6.1) . Let S := X e ∼ E µ ( e ) X q ( q,a )=1 ψ (cid:16) qQ (cid:17) X d ∼ D ( d,a )=1 X r ,r ∼ R ( r r ,ae )=1 η q,d,r η q,d,r × X n ,n ∼ Nn ≡ n (mod qde )( n ,qder )=1( n ,qder )=1 α n α n X mmn ≡ a (mod qdr ) mn ≡ a (mod qdr ) ψ (cid:16) mM (cid:17) . Let C = C ( A, B ) be sufficiently large in terms of A and B , and let (14.3) QDE (log x ) C < N. Then we have S = S MT + O ( M | E | ) + O A,B (cid:16) M N QD (log x ) A (cid:17) , where S MT is as given by Lemma 14.1, H := QDR M log x and where E := X e ∼ E µ ( e ) X q ( q,a )=1 ψ (cid:16) qQ (cid:17) X d ∼ D ( d,a )=1 X r ,r ∼ R ( r ,ar )=1( r ,aqdr )=1 η q,d,r η q,d,r qdr r × X n ,n ∼ Nn ≡ n (mod qde )( n ,n eqdr )=1( n ,n eqdr )=1 | n − n |≥ N/ (log x ) C α n α n X ≤| h |≤ H b ψ (cid:16) hMqdr r (cid:17) e (cid:16) ahn r qdr + ahn qdr r (cid:17) . Proof. Again, for notational simplicity, we let ℓ = qd throughout the proof. Firstwe wish to modify the the summation conditions slightly. We first remove termswith | α n α n η q,d,r η q,d,r | ≥ (log x ) C or τ ( mn − a ) τ ( mn − a ) ≥ (log x ) C . Theseterms contribute a total ≪ X e ∼ E X ℓ ≍ QD X n ,n ∼ Nn ≡ n (mod ℓe ) X m ∼ Mmn ≡ a (mod ℓ ) mn ≡ a (mod ℓ ) X r | mn − a × X r | mn − a τ ( mn − a ) τ ( mn − a )( τ ( n ) τ ( n ) τ ( ℓ ) τ ( r ) τ ( r )) B (log x ) C ≪ X n ∼ N X m ∼ M X ℓ ≍ QDℓ | mn − a τ ( ℓ ) B τ ( mn − a ) B +2 τ ( n ) B (log x ) C X e ∼ E X n ∼ Nn ≡ n (mod ℓe ) ≪ E (cid:16) NQDE + 1 (cid:17) X n ∼ N X m ∼ M τ ( mn − a ) B +3 τ ( n ) B (log x ) C ≪ M N (log x ) O B (1) − C QD + EM N (log x ) O B (1) − C . Here we used Cauchy-Schwarz and the symmetry of n and n to pass to thesecond line, and used Lemma 8.7 in the final line. Thus, if we fix C = C ( A, B )to be sufficiently large in terms of A and B , by (14.3) these terms contribute O A,B ( M N / ( QD log A x )). We denote the restriction to | α n α n η q,d,r η q,d,r | ≤ (log x ) C and τ ( mn − a ) τ ( mn − a ) ≤ (log x ) C by P ∗ .Since β n and η q,d,r are supported on integers with P − ( n ) , P − ( r ) ≥ z , we expectthat n and n should be typically coprime to one another, and similarly r and r and qd and r . Indeed, the contribution with ( r , r ) > S is X f>z X e ∼ E X r ,r ∼ Rf | r , f | r X ℓ ≍ QD X n ,n ∼ Nn ≡ n (mod ℓe ) X ∗ m ≪ Mmn ≡ a (mod ℓr ) mn ≡ a (mod ℓr ) (log x ) C ≪ X n ∼ N X ∗ m ∼ M X f>z f | mn − a X e ∼ E X ℓ ≍ QDℓ | mn − a X ∗ n ∼ Nn ≡ n (mod ℓ [ f,e ]) X r ′ | mn − a X r ′ | mn − a (log x ) C ≪ X n ∼ N X ∗ m ∼ M X f>z f | mn − a X e ∼ E (cid:16) ( e, f ) NQDEz + 1 (cid:17) (log x ) C ≪ M N (cid:16) NQDz + E (cid:17) (log x ) C ≪ M N (log x ) C QDz + EM N (log x ) C . Thus, assuming (14.3) holds with C > C +2 A , this is O A,B ( M N / ( QD log A x )),and so negligible. Similarly, we restrict to ( n , n ) = 1. The contribution to S from terms with ( n , n ) > ≪ X f>z X e ∼ E X n ∼ Nf | n X ∗ m ∼ M X ℓ ≍ QDℓ | mn − a X ∗ n ∼ Nn ≡ n (mod ℓ [ e,f ]) X r | mn − a X r | mn − a (log x ) C ≪ X f>z X n ∼ Nf | n X m ≪ M X e ∼ E (cid:16) ( e, f ) NQDEz + 1 (cid:17) (log x ) C ≪ M N (log x ) C +2 QDz + EM N (log x ) C +2 . Similarly, we restrict to ( qd, r ) = 1. Terms with ( qd, r ) > ≪ X e ∼ E X n ∼ N X ∗ m ∼ M X ℓ ≍ QDℓ | mn − a ( ℓ,a )=1 X f>z f | ℓ X ∗ n ∼ Nmn ≡ a (mod ℓf ) n ≡ n (mod ℓe ) X r | mn − a X r ′ | mn − a (log x ) C ≪ X n ∼ N X ∗ m ∼ M X f | mn − a X e ∼ E (cid:16) N ( e, f ) QDEz + 1 (cid:17) (log x ) C ≪ M N (log x ) C QDz + EM N (log x ) C . RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 51 Similarly, we restrict to | n − n | ≥ N/ (log x ) C . Terms with | n − n | ≤ N/ (log x ) C contribute a total ≪ X e ∼ E X n ∼ N X ∗ m ∼ M X ℓ ≍ QDℓ | mn − a X ∗ n ∼ Nn ≡ n (mod ℓe ) | n − n |≤ N/ (log x ) C X r | mn − a X r | mn − a (log x ) C ≪ EN M (log x ) C (cid:16) NQDE (log x ) C + 1 (cid:17) (log x ) C ≪ M N (log x ) C QD (log x ) C + EM N (log x ) C . Finally, we remove the condition ( e, r r ) = 1. Terms with ( e , r ) > X f>z X e ∼ Ef | e X n ∼ N X ∗ m ∼ M X ℓ ≍ QDℓ | mn − a X n ∼ Nn ≡ n (mod ℓe ) mn ≡ a (mod ℓf ) X r | mn − a X r ′ | mn − a (log x ) C ≪ X ∗ m ∼ M X n ∼ N X ff | mn − a Ez (cid:16) NQDE + 1 (cid:17) (log x ) C ≪ M N (log x ) C QDz + EM N (log x ) C z . Provided C > C + 2 A + 2 all of these are negligible. After making these restric-tions, we re-insert terms with τ ( mn − a ) τ ( mn a ) ≥ (log x ) C and | α n α n η q,d,r η q,d,r | ≥ (log x ) C , which we have already seen is a negligible contribution. Thus, assuming(14.3) holds with C > C + 2 A + 2, we see that(14.4) S = S + O A,B (cid:16) M N QD (log x ) A (cid:17) , where S := X e ∼ E µ ( e ) X q ( q,a )=1 ψ (cid:16) qQ (cid:17) X d ∼ D ( d,a )=1 X r ,r ∼ R ( r ,ar )=1( r ,aqdr )=1 η q,d,r η q,d,r X n ,n ∼ N ( n ,n qder )=1( n ,n qder )=1 | n − n |≥ N/ (log x ) C n ≡ n (mod qde ) α n α n × X mmn ≡ a (mod qdr ) mn ≡ a (mod qdr ) ψ (cid:16) mM (cid:17) . We see since n ≡ n (mod ℓ ), then m is fixed to lie in a certain residue class b (mod ℓr r ) where b ≡ an (mod ℓr ) and b ≡ an (mod r ) by the ChineseRemainder Theorem and the fact that ( ℓr , r ) = 1. In particular, we see that bℓr r = an r ℓr + an ℓr r (mod 1) . Thus, by Lemma 13.4, we have X m ≡ b (mod ℓr r ) ψ (cid:16) mM (cid:17) = Mℓr r b ψ (0)+ Mℓr r X ≤| h |≤ H b ψ (cid:16) hMℓr r (cid:17) e (cid:16) ah (cid:16) n r ℓr + n ℓr r (cid:17)(cid:17) + O ( x − ) , where H := QDR M log x. The final term clearly makes a negligible contribution. The contribution to S from the first term is M b ψ (0) X e ∼ E µ ( e ) X q ( q,a )=1 ψ (cid:16) qQ (cid:17) X d ∼ D ( d,a )=1 X r ,r ∼ R ( r ,ar )=1( r ,aqdr )=1 η q,d,r η q,d,r qdr r X n ,n ∼ Nn ≡ n (mod qde )( n ,qder )=1( n ,qder )=1( n ,n )=1 | n − n |≥ N/ (log x ) C α n α n . We multiply the summand above by a factor φ ( qde ) φ ( qdr r ) φ ( qd ) φ ( qder ) φ ( qder ) = Y p | ( r r ,e ) p ∤ qd (cid:16) p − p (cid:17) Y p | ( r ,r ) p ∤ qde (cid:16) pp − (cid:17) = 1 + O (cid:16) log xz (cid:17) , where we have used the fact that P − ( r ) ≥ z from the support of η . This introducesan error term which contributes ≪ M log xz X e ∼ E X ℓ ≍ QD X r ,r ∼ R ( τ ( r ) τ ( r ) τ ( ℓ ) ) B QDR X n ,n ∼ Nn ≡ n (mod ℓe ) | n − n |≥ N/ (log x ) C ( τ ( n ) τ ( n )) B ≪ M N (log x ) O B (1) QDz + EM N (log x ) O B (1) z . This is negligible assuming (14.3) holds with C sufficiently large in terms of B and A . Thus we see that(14.5) S = S + O ( M | E | ) + O A,B (cid:16) M N QD (log x ) A (cid:17) , where E is is given by the statement of the lemma, and where S := M b ψ (0) X e ∼ E µ ( e ) X q ( q,a )=1 ψ (cid:16) qQ (cid:17) X d ∼ D ( d,a )=1 X r ,r ∼ R ( r ,ar )=1( r ,aqdr )=1 × η q,d,r η q,d,r φ ( qde ) φ ( qdr r ) φ ( qd ) φ ( qder ) φ ( qder ) qdr r X n ,n ∼ Nn ≡ n (mod qde )( n ,n qder )=1( n ,n qder )=1 α n α n . RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 53 We see that we may drop the conditions ( r n , r n ) = 1 and ( r , ℓr ) = 1 in S at the cost of an error of size X z 1, which we have just seen are negligible. In particular,this implies that(14.6) S = S MT + E + O A (cid:16) M N QD (log x ) A (cid:17) , where E is given by M b ψ (0) X e ∼ E µ ( e ) X q ( q,a )=1 ψ (cid:16) qQ (cid:17) X d ∼ D ( d,a )=1 X r ,r ∼ R ( r r ,a )=1 η q,d,r η q,d,r φ ( qd ) φ ( qdr r ) φ ( qdr ) φ ( qdr ) ℓr r X b (mod qde )( b,qde )=1 × X n ,n ∼ N α n α n (cid:16) n ≡ b (mod qde ) − ( n ,qde )=1 φ ( qde ) (cid:17)(cid:16) n ≡ b (mod qde ) − ( n ,qde )=1 φ ( qde ) (cid:17) . Trivially bounding the r , r summation and letting s = qde , we see that thissimplifies to give the bound E ≪ M (log x ) O B (1) QD X s ≤ QDE τ ( s ) B X b (mod s )( b,s )=1 (cid:12)(cid:12)(cid:12) X n ∼ N α n (cid:16) n ≡ b (mod s ) − ( n,s )=1 φ ( s ) (cid:17)(cid:12)(cid:12)(cid:12) . Since α n satisfies the Siegel-Walfisz condition (6.1), by Lemma 13.8 we see that(using (14.3)) we have(14.7) E ≪ A,B M N QD (log x ) A . Putting together (14.4), (14.5), (14.6) and (14.7) then gives the result. (cid:3) Proposition 14.4 (Reduction to exponential sums) . Let α n , β m , γ q,d , λ q,d,r be com-plex sequences with | α n | , | β n | ≤ τ ( n ) B and | γ q,d | ≤ τ ( qd ) B and | λ q,d,r | ≤ τ ( qdr ) B .Let α n and λ q,d,r be supported on integers with P − ( n ) ≥ z and P − ( r ) ≥ z , andlet α n satisfy the Siegel-Walfisz condition (6.1) . Let S := X d ∼ D ( d,a )=1 X q ∼ Q ( q,a )=1 X r ∼ R ( r,a )=1 λ q,d,r γ q,d X m ∼ M β m X n ∼ N α n (cid:16) mn ≡ a (mod qrd ) − ( mn,qrd )=1 φ ( qrd ) (cid:17) . Let A > , ≤ E ≤ x and C = C ( A, B ) be sufficiently large in terms of A, B ,and let N, M satisfy N > QDE (log x ) C , M > (log x ) C . Then we have | S | ≪ A,B x (log x ) A + M D / Q / (log x ) O B (1) (cid:16) | E | / + | E | / (cid:17) , where E := X e ∼ E µ ( e ) X q ( q,a )=1 X d ∼ D ( d,a )=1 X r ,r ∼ R ( r r ,a )=1 ψ (cid:16) qQ (cid:17) λ q,d,r λ q,d,r φ ( qder ) qdr × X n ,n ∼ N ( n ,qder )=1( n ,qder )=1 α n α n X ≤| h |≤ H b ψ (cid:16) hMqdr (cid:17) e (cid:16) ahn qdr (cid:17) , E := X e ∼ E µ ( e ) X q ( q,a )=1 ψ (cid:16) qQ (cid:17) X d ∼ D ( d,a )=1 X r ,r ∼ R ( r ,ar )=1( r ,aqdr )=1 λ q,d,r λ q,d,r qdr r × X n ,n ∼ Nn ≡ n (mod qde )( n ,n eqdr )=1( n ,n eqdr )=1 | n − n |≥ N/ (log x ) C α n α n X ≤| h |≤ H b ψ (cid:16) hMqdr r (cid:17) e (cid:16) ahn r qdr + ahn qdr r (cid:17) ,H := QDRM log x,H := QDR M log x. Remark. To avoid confusion, we emphasize to the reader that in the above propo-sition (and elsewhere in the paper), e is the integer in the outer summation, exceptfor when we use the function e ( x ) = e πix . The distinction between the two shouldbe clear from the context, and in most situations when this is used the variable e will just be the constant 1.Proof. We first do some slightly artifical looking manipulations (which will be vitallater on), where we split S according to residue classes for additional moduli e ∼ E .(This corresponds to the variable labelled c in Section 3.4.) RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 55 Given an integer e ∼ E , since α n is supported on P − ( n ) > z , we see that thecontribution from ( e, n ) = 1 to S is ≪ X f>z f | e X d ∼ D X q ∼ Q X r ∼ R X m ∼ M X n ∼ Nf | n ( nm,qdr )=1 τ ( qdrnm ) B (cid:16) mn ≡ a (mod qdr ) + 1 φ ( qdr ) (cid:17) ≪ X f>z f | e DQR xτ ( f ) O B (1) QDRf (log x ) O B (1) ≪ x (log x ) O B z / X f | e τ ( f ) O B (1) f / ≪ x (log x ) O B z / exp( O B ((log x ) / )) . Similarly, the contribution with ( e, r ) = 1 is ≪ X f>z f | e X d ∼ D X q ∼ Q X r ∼ Rf | r X m ∼ M X n ∼ N τ ( qdrnm ) B (cid:16) mn ≡ a (mod qdr ) + 1 φ ( qdr ) (cid:17) ≪ x (log x ) O B X f>z f | e τ ( f ) O B (1) f ≪ x (log x ) O B z / exp( O B ((log x ) / ))Since z = x / (log log x ) , we see that these error terms are negligible, and so we mayinsert the conditions ( n, e ) = 1 and ( r, e ) = 1 at a negligible cost. We see that allterms are supported only on ( m, qd ) = 1, and so we also insert this explicitly. Wenow split the terms according to the residue class of n (mod qde ). Together thisgives S = X d ∼ D ( d,a )=1 X q ∼ Q ( q,a )=1 X r ∼ R ( r,ae )=1 λ q,d,r γ q,d X m ∼ M ( m,qd )=1 β m X b (mod qde ) b ≡ am (mod qd )( b,qde )=1 × X n ∼ N ( n,e )=1 α n (cid:16) mn ≡ a (mod qrd ) n ≡ b (mod qde ) − ( mn,qrd )=1 φ ( qrde ) (cid:17) + O A (cid:16) x (log x ) A (cid:17) . (Here we used the fact that there are precisely φ ( qde ) /φ ( qd ) choices of b in thesummation.) We now average this expression over squarefree e ∼ E . This gives S = 1 E X e ∼ E µ ( e ) X d ∼ D ( d,a )=1 X q ∼ Q ( q,a )=1 X r ∼ R ( r,ae )=1 λ q,d,r γ q,d X m ∼ M β m X b (mod qde ) b ≡ am (mod qd )( b,qde )=1 × X n ∼ N ( n,e )=1 α n (cid:16) mn ≡ a (mod qrd ) n ≡ b (mod qde ) − ( mn,qrd )=1 φ ( qrde ) (cid:17) + O A (cid:16) x (log x ) A (cid:17) , where E := X e ∼ E µ ( e ) ≍ E. We now apply Cauchy-Schwarz in the e , b , m , q and d variables, which allows usto eliminate the β m and γ q,d coefficients. This gives | S | ≤ E (cid:16)X e ∼ E X q ∼ Q X d ∼ D X m ∼ M ( m,qd )=1 X b (mod qde ) b ≡ am (mod qd ) | γ q,d | | β m | (cid:17) S ′ + O A (cid:16) x (log x ) A (cid:17) ≪ M QD (log x ) O B (1) S ′ + O A (cid:16) x (log x ) A (cid:17) , where S ′ := X e ∼ E µ ( e ) X q ∼ Q ( q,a )=1 X d ∼ D ( d,a )=1 X m ∼ M X b (mod qde ) b ≡ am (mod qd )( b,qde )=1 × (cid:12)(cid:12)(cid:12) X r ∼ R ( r,ae )=1 λ q,d,r X n ∼ N ( n,e )=1 α n (cid:16) mn ≡ a (mod qdr ) n ≡ b (mod qde ) − ( mn,qdr )=1 φ ( qdre ) (cid:17)(cid:12)(cid:12)(cid:12) . We now extend the m and q summations using ψ for an upper bound, and thenopen out the square. This gives S ′ ≤ X q ( q,a )=1 X d ∼ D ( d,a )=1 X m ψ (cid:16) qQ (cid:17) ψ (cid:16) mM (cid:17) X e ∼ E µ ( e ) X b (mod qde ) b ≡ am (mod qd )( b,qde )=1 × (cid:12)(cid:12)(cid:12) X r ∼ R ( r,ae )=1 λ q,d,r X n ∼ N ( n,e )=1 α n (cid:16) mn ≡ a (mod qdr ) n ≡ b (mod qde ) − ( mn,qdr )=1 φ ( qdre ) (cid:17)(cid:12)(cid:12)(cid:12) = S − ℜ ( S ) + S , (14.8)where, after performing the b summation, we see that S , S , S are as givenby Lemma 14.1, 14.2 and 14.3 with η q,d,r := λ q,d,r . These lemmas then give theresult. (cid:3) RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 57 Lemma 14.5 (Simplification of exponential sum) . Let N, M, Q, R ≤ x with N M ≍ x and QR < x / , (14.9) QR < M x − ǫ . (14.10) Let λ q,r and α n be complex sequences supported on P − ( n ) , P − ( r ) ≥ z with | λ q,r | ≤ τ ( qr ) B and | α n | ≤ τ ( n ) B . Let H := QR M log x and let E := X ( q,a )=1 ψ (cid:16) qQ (cid:17) X r ,r ∼ R ( r ,ar )=1( r ,aqr )=1 λ q,r λ q,r qr r X n ,n ∼ Nn ≡ n (mod q )( n ,n qr )=1( n ,n qr )=1 | n − n |≥ N/ (log x ) C α n α n × X ≤| h |≤ H b ψ (cid:16) hMqr r (cid:17) e (cid:16) ahn r qr + ahn qr r (cid:17) . Then we have (uniformly in C ) E ≪ B exp((log log x ) ) sup H ′ ≤ HQ ′ ≤ QR ,R ≤ R | E ′ | + N Qx ǫ , where E ′ = X Q ≤ q ≤ Q ′ ( q,a )=1 X R ≤ r ≤ R R ≤ r ≤ R ( r ar )=1( r ,aqr )=1 λ q,r λ q,r qr r X n ,n ∼ Nn ≡ n (mod q )( n ,qr n )=1( n ,qr n )=1( n r ,n ) ∈N| n − n |≥ N/ (log x ) C α n α n X ≤| h |≤ H ′ e (cid:16) ahn qr ( n − n ) n r (cid:17) , and N is a set with the property that if ( a, b ) ∈ N and ( a ′ , b ′ ) ∈ N then we have gcd( a, b ′ ) = gcd( a ′ , b ) = 1 .Proof. We first wish to simplify the exponential term in the summand. By Lemma13.1, we have ahn r qr = − ahqr n r + ahqr r n (mod 1) . Since | h | ≤ H = ( QR log x ) /M , the final fraction is of size O (log x/x ), and so seethat e (cid:16) ahn r qr + ahn qr r (cid:17) = e (cid:16) ahn qr ( n − n ) n r (cid:17) + O (cid:16) log xx (cid:17) . The error term above contributes to E a total ≪ N (cid:16) NQ + 1 (cid:17) QR (log x ) O B (1) M x ≪ N Q (log x ) O B (1) (cid:16) QR M x + Q R x (cid:17) . Therefore if (14.9) and (14.10) hold, both terms in parentheses are ≪ B x − ǫ . Thuswe see that E = E + O B (cid:16) N Qx ǫ (cid:17) , where E := X ( q,a )=1 ψ (cid:16) qQ (cid:17) X r ,r ∼ R ( r ,ar )=1( r ,aqr )=1 λ q,r λ q,r qr r X n ,n ∼ Nn ≡ n (mod q )( n ,n qr )=1( n ,n qr )=1 | n − n |≥ N/ (log x ) C α n α n × X ≤| h |≤ H b ψ (cid:16) hMqr r (cid:17) e (cid:16) ahn qr ( n − n ) n r (cid:17) . We note that since b ψ ( j )0 ( ξ ) ≪ j,k | ξ | − k and ψ is supported on [1 / , / ∂ k + k + k + k ∂q k ∂r k ∂r k ∂h k ψ (cid:16) qQ (cid:17) b ψ (cid:16) hMqr r (cid:17) ≪ k ,k ,k ,k q k r k r k h k . Thus we may remove the b ψ term via partial summation, which shows that E ≪ sup H ′ ≤ HQ ′ ≤ QR ,R ≤ R | E | , where E := X Q ≤ q ≤ Q ′ ( q,a )=1 X R ≤ r ≤ R R ≤ r ≤ R ( r ,ar )=1( r ,aqr )=1 λ q,r λ q,r qr r X n ,n ∼ Nn ≡ n (mod q )( n ,qr n )=1( n ,qr n )=1 | n − n |≥ N/ (log x ) C α n α n ′ X ≤| h |≤ H ′ e (cid:16) ahn qr ( n − n ) n r (cid:17) . We now apply Lemma 13.2, recalling that α n and λ q,r are supported on integers q, n, r with P − ( n ) , P − ( r ) ≥ z , and so n and r have at most (log log x ) primefactors. This gives us ≪ exp((log log x ) ) different sets N , N , · · · ⊆ Z whichcover all possible values of pairs ( n r , n ) such that if ( n r , n ) and ( n ′ r ′ , n ′ ) arein the same set N j then we have gcd( n r , n ′ ) = gcd( n ′ r ′ , n ) = 1. Thus, takingthe worst such set N , we find E ≪ exp((log log x ) ) sup N E ′ , with E ′ given by the lemma. Putting everything together then gives the result. (cid:3) RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 59 Fouvry-style estimates near x / In this section we establish Proposition 12.1. The arguments here are a generaliza-tion of [14], making use of the fact that we consider only moduli with a convenientlysized prime factor to extend the results to a wider region. The case Q = 1 is pre-cisely the argument of [14] (or [3, Theorem 3]). The critical case for these estimatesis handling convolutions when one factor is of size x / and the other of size x / or when one factor is of size x / and the other of size x / . Lemma 15.1 (Deshouillers–Iwaniec) . Let b n,r,s be a 1-bounded sequence and R, S, N, D, C ≪ x O (1) . Let g ( c, d ) = g ( c/C, d/D ) where g is a smooth function supported on [1 / , / × [1 / , / . Then we have X r ∼ R X s ∼ S ( r,s )=1 X n ∼ N b n,r,s X d ∼ D X c ∼ C ( rd,sc )=1 g ( c, d ) e (cid:16) ndrcs (cid:17) ≪ g x ǫ (cid:16)X r ∼ R X s ∼ S X n ∼ N | b n,r,s | (cid:17) / J . where J = CS ( RS + N )( C + DR ) + C DS p ( RS + N ) R + D N R. Proof. This is [9, Theorem 12] (correcting a minor typo in the last term of J which is written as D N R/S ). (cid:3) Lemma 15.2 (Fouvry-style exponential sum estimate) . Let N, Q, R satisfy N Q R < x − ǫ , (15.1) N Q R < x − ǫ , (15.2) N Q R < x − ǫ , (15.3) N Q R < x / − ǫ , (15.4) QR < x / − ǫ N / . (15.5) Let H = ( QR log x ) /M and λ q,r and α n be 1-bounded complex sequences. Let F := X ( q,a )=1 ψ (cid:16) qQ (cid:17) X r ,r ∼ R ( r ,ar )=1( r ,aqr )=1 λ q,r λ q,r qr r X n ,n ∼ Nn ≡ n (mod q )( n ,n qr )=1( n ,n qr )=1 | n − n |≥ N/ (log x ) C α n α n × X ≤| h |≤ H b ψ (cid:16) hMqr r (cid:17) e (cid:16) ahn r qr + ahn qr r (cid:17) . Then we have F ≪ N Qx ǫ . Proof. We first apply Lemma 14.5 to F . This shows that F ≪ N Qx ǫ + exp((log log x ) ) sup H ′ ≤ HQ ′ ≤ QR ,R ≤ R | F | , with F as given by E ′ in Lemma 14.5. We now rewrite the congruence n ≡ n (mod q ) by n = n + kq for some N/ ( q log C x ) ≤ | k | ≤ K where K := NQ . Thus we see that F = X Q ≤ q ≤ Q ′ ( q,a )=1 X R ≤ r ≤ R R ≤ r ≤ R ( r ,ar )=1( r ,aqr )=1 λ q,r λ q,r qr r X n ∼ N ( n,qr )=1 X N/ ( q log C x ) ≤| k |≤ Kn + kq ∼ N ( n + kq,qr n )=1( nr ,n + kq ) ∈N α n α n + kq × X ≤| h |≤ H ′ e (cid:16) − ahk ( n + kq ) r nr (cid:17) . We wish to show F ≪ N /Qx ǫ . We apply Cauchy-Schwarz in the q, r , r vari-ables to eliminate the λ q,r coefficients, giving F ≪ | F | / Q / R , where F := X q ∼ Q X r ,r ∼ R ( r ,r )=1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ∼ N ( n,qr )=1 X N/ ( q log C x ) ≤| k |≤ Kn + kq ∼ N ( n + kq,qnr )=1( nr ,n + kq ) ∈N X ≤| h |≤ H ′ c n,k,q,h e (cid:16) − ahk ( n + kq ) r nr (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , for some 1-bounded coefficients c n,k,q,h . We wish to show F ≪ R N /Qx ǫ . (Herewe have extended the summation of q, r , r slightly for an upper bound.)We insert smooth majorants for r and r , extend the summation for an upperbound, then expand the square and swap the order of summation. After droppingsome summation conditions on the outer variables for an upper bound, this leavesus with F ≤ X n ,n ∼ N X | k | , | k |≤ Kn + k q ∼ Nn + k q ∼ N X ≤| h | , | h |≤ H ′ X q ∼ Q ( n + k q,n n )=1( n + k q,n n )=1 (cid:12)(cid:12)(cid:12) X r ,r ( r ,n n r )=1( r ,r s )=1 g ( r , r ) e (cid:16) ℓsr n n r (cid:17)(cid:12)(cid:12)(cid:12) , where g ( r , r ) := ψ (cid:16) r R (cid:17) ψ (cid:16) r R (cid:17) ,ℓ := a ( h k n ( n + k q ) − h k n ( n + k q )) ,s := ( n + k q )( n + k q ) . Here we have crucially made use of the fact that ( n r , n + k q ) , ( n r , n + k q ) ∈ N and so we may assume gcd( n , n + k q ) = gcd( n , n + k q ) = 1 to give theconditions in the q -summation.We split our upper bound for F according to whether ℓ = 0 or not: F ≤ F ℓ =0 + F ℓ =0 . RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 61 We first consider F ℓ =0 , the contribution from the ‘diagonal’ terms with ℓ = 0.Since ( n , n + k q ) = 1 we have ( k , n + k q ) = 1 and so k | h k n for theseterms. Thus F ℓ =0 ≪ X q ∼ Qr ,r ∼ R X | h |≤ H | k |≤ Kn ∼ N X k | h k n X h n | h k n ( n + k q ) ≪ (log x ) O (1) QR HKN ≪ (log x ) O (1) QR N M . This gives F ℓ =0 ≪ R N /Qx ǫ provided M > x ǫ Q R N . Recalling M N ≍ x , this is satisfied if(15.6) QR < x / − ǫ N / . We now consider the terms with ℓ = 0. We let t = n n and put each of ℓ , t , s intoone of O (log x ) dyadic ranges. Thus(15.7) F ℓ =0 ≪ (log x ) sup N ≪ S,T ≪ N L ≪ R N (log x ) /M X s ∼ S X t ∼ T X ℓ ∼ L b ℓ,s,t X r ,r ( sr ,r t )=1 g ( r , r ) e (cid:16) ℓsr r t (cid:17) , where b ℓ,s,t are complex coefficients satisfying | b ℓ,s,t | ≤ X q ∼ Q X | k | , | k |≤ K X ≤| h | , | h |≤ H ′ X n ,n ∼ Nℓ = a ( h k n ( n + k q ) − h k n ( n + k q )) s =( n + k q )( n + k q ) t = n n . We can now apply Lemma 15.1 to bound the sum (15.7). This gives F ℓ =0 ≪ x ǫ (cid:16) RT ( ST + L )( R + RS ) + R T p ( ST + L ) S + R LS (cid:17) / k b k ≪ x ǫ RN (cid:16) N + N R M (cid:17) ( R + RN )+ R N r(cid:16) N + N R M (cid:17) N + R N R M N ! / k b k . Since M N ≍ x , this simplifies to F ℓ =0 ≪ x ǫ (cid:16) R N + R N x + R N + R N x / (cid:17) / k b k . (15.8)We first consider k b k . We see that k b k ≤ X ℓ ∼ L X s ∼ S X t ∼ T ( s,t )=1 (cid:16) X n n = tn ,n ∼ N X | k | , | k |≤ Kq ∼ Qs =( n + k q )( n + k q ) C (cid:17) , where C = C ( k , k , q, n , n , ℓ ) is the set C = { ( h , h ) ∈ [1 , H ] : ℓ = a ( h k n ( n + k q ) − h k n ( n + k q )) } . There are at most τ ( t ) = x o (1) choices of n , n given t . There are at most τ ( s ) = x o (1) choices of n + k q and n + k q given s . There are then at most τ ( s − n ) τ ( s − n ) = x o (1) choices of k , k , q given n + k q and n + k q and n , n . Thus thereare at most x o (1) choices of n , n , k , k , q given s and t . Moreover, since ( s, t ) = 1we have ( n , k ) = ( n , k ) = 1. Thus, by Cauchy-Schwarz, we see that k b k ≪ x o (1) X ℓ ∼ L X s ∼ S X t ∼ T ( s,t )=1 X n n = tn ,n ∼ N X q ∼ Q X | k | , | k |≤ Ks =( n + k q )( n + k − q ) C ≪ x o (1) X n ,n ,k ,k ,q X h ,h ,h ,h ( h − h ) k n ( n + k q )=( h − h ) k n ( n + k q )( n ,k )=( n ,k )=1 . Here we have suppressed the ranges for the variables in the final line for convenience.We see that if h − h = 0, then we must have h − h = 0, and so the totalcontribution from terms with h − h = 0 is ≪ x o (1) H N K Q ≪ x o (1) Q R M N N Q Q ≪ x o (1) QR N x . (15.9)If instead h − h = 0, then we first fix a choice of h , h , n , k and q , for whichthere are O ( H N KQ ) choices. Since ( k , n ) = 1, we have that n ( n + k q ) | ( h − h ) n ( n + k q ), so there are O ( x o (1) ) choices of n , k . Given such a choice of n , k , we see that ( h − h ) is uniquely determined, and so there are O ( H ) choicesof h and h . Thus the total contribution from terms with h − h = 0 is(15.10) ≪ x o (1) H N KQ ≪ x o (1) QR N x (cid:16) Q R xN (cid:17) . But we have Q R < x − ǫ N by (15.6), so we find that the bound (15.9) is largerthan the bound (15.10). Thus(15.11) k b k ≪ x ǫ QR N x . Putting together (15.11) and (15.8), we find that F ℓ =0 ≪ x ǫ (cid:16) R N + R N x + R N + R N x / (cid:17) / (cid:16) QR N x (cid:17) / ≪ x ǫ (cid:16) Q / R N x + Q / R N x / + Q / R / N / x + Q / R N / x / (cid:17) . Thus we have F ℓ =0 ≪ R N / ( Qx ǫ ) provided we have N R Q < x − ǫ , (15.12) N R Q < x − ǫ , (15.13) N R Q < x − ǫ , (15.14) N R Q < x / − ǫ . (15.15)This gives the result. (cid:3) RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 63 Proof of Proposition 12.1. First we note that by Lemma 8.7 the set of n, m withmax( | α n | , | β m | ) ≥ (log x ) C has size ≪ x (log x ) O B (1) − C , so by Lemma 8.9 theseterms contribute negligibly if C = C ( A, B ) is large enough. Thus, after dividingthrough by (log x ) C and replacing A with A + 2 C , it suffices to show the resultwhen α n β m are 1-bounded ( α n still satisfies (6.1) by Lemma 13.7.)We factor q = rd where P − ( r ) > z ≥ P + ( d ) into parts with large and smallprime factors. By putting these in dyadic intervals, we see that it suffices to showfor every A > DR ≍ Q that X q ∼ Q ( q ,a )=1 X r ∼ RP − ( d ) >z ( r,a )=1 X d ∼ DP + ( d ) ≤ z | ∆( q rd ) | ≪ A x (log x ) A . By Lemma 13.10 we have the result unless D ≤ y = x / log log x , and so we mayassume that R = Q x − o (1) . We let q = q d and Q ≍ Q D , and see that it sufficesto show X q ∼ Q ( q,a )=1 τ ( q ) X r ∼ RP − ( d ) >z ( r,a )=1 | ∆( qr ) | ≪ A x (log x ) A . We insert coefficients c q,r to remove the absolute values, and absorb the conditions P − ( r ) > z , q ∼ Q into the definition of c q,r . We now see that we have a sum of thetype considered in Proposition 14.4, which we can apply provided N > Q (log x ) C .By the assumptions of the proposition we also have that N Q Q < x − ǫ , so wehave H = ( QR log x ) /M < E of Proposition 14.4 vanishes.Therefore, by Proposition 14.4, it suffices to show that E ≪ N Qx ǫ , where E := X ( q,a )=1 ψ (cid:16) qQ (cid:17) X r ,r ∼ R ( r ,ar )=1( r ,aqr )=1 λ q,r λ q,r qr r X n ,n ∼ Nn ≡ n (mod q )( n ,n qr )=1( n ,n qr )=1 | n − n |≥ N/ (log x ) C α n α n × X ≤| h |≤ H b ψ (cid:16) hMqr r (cid:17) e (cid:16) ahn r qr + ahn qr r (cid:17) , where Q = Q x o (1) , R = Q x − o (1) and λ q,r = c q,r are 1-bounded coefficientssupported on P − ( r ) ≥ z . We now see that Lemma 15.2 gives the result providedwe have Q log C x < N, N R Q < x − ǫ , N R Q < x − ǫ ,N R Q < x − ǫ , N R Q < x / − ǫ , QR < x / − ǫ N / . The result also follows from the Bombieri-Vinogradov Theorem if QR < x / − ǫ , sowe may assume that QR > x / − ǫ . We then see that the fourth condition impliesthat N < x / , so the third condition follows from the fifth, and may be dropped. Recalling that Q = Q x o (1) , and R = Q x − o (1) and noting the conditions on Q , Q then gives the result. (cid:3) RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 65 Small divisor estimates near x / In this section we establish Proposition 12.2. This is a slightly different arrange-ment of the sums appearing, but has similarities with the work of Fouvry [14]. Itultimately relies on the Weil bound, but is only effective when N is a relativelysmall power of x . This is important in handling products of three large smooth fac-tors and one small factor. Roughly similar results via an alternative method wereobtained by Fouvry and Radziwi l l [21], but their results (which hold regardless offactorization properties of the moduli) do not extend to moduli which are largeenough for our applications. The critical case is for convolutions when one factoris of size x / and the other of size x / . Lemma 16.1 (Kloosterman sum bound) . X n ( n,q )=1 ψ (cid:16) nN (cid:17) e (cid:16) anq (cid:17) ≪ q o (1) (cid:16) q / + Nq ( a, q ) (cid:17) . Proof. This is the classic Ramanujan-Weil bound combined with completion ofsums - it follows immediately from Lemma 13.5, noting that the first term involvesa Ramanujan sum so is of size N ( a, q ) /q and the second term involves a Kloostermansum, which is bounded by q / o (1) ( h, q ) / by Lemma 13.3. (cid:3) Lemma 16.2 (Small divisor exponential sum estimate) . Let M, N, Q, R, D ≥ satisfy D ≤ x o (1) , N M ≍ x and N Q R < x − ǫ ,Q R < x − ǫ N, and let H = ( DQ R log x ) /M . Let η d,q,r be a 1-bounded complex sequence withsupported on qr having no prime factors less than z := x / (log log x ) , and let α n bea 1-bounded sequence. Let D := X ( d,a )=1 ψ (cid:16) dD (cid:17) X q ,q ∼ Q ( q ,aq )=1( q ,adq )=1 X r ,r ∼ R ( r ,aq r )=1( r ,adq r )=1 η d,q ,r η d,q ,r dq q r r × X n ,n ∼ Nn ≡ n (mod d )( n ,n dq r )=1( n ,n dq r )=1 | n − n |≥ N/ (log x ) C α n α n X ≤| h |≤ H b ψ (cid:16) hMdq q r r (cid:17) e (cid:16) ahn q r dq r + ahn dq r q r (cid:17) . Then we have for every A > (uniformly in C > ) D ≪ A N D (log x ) A . Proof. We first note that D is a special case of the sort of sum considered in Lemma14.5 when the coefficients λ q,r are of the form λ d,b = X b = qrq ∼ Qr ∼ R η d,q,r . Thus we may apply Lemma 14.5. Taking the worst d ∼ D (and removing the 1 /d factor), this gives D ≪ exp((log log x ) ) sup H ′ ≤ Hd ≤ DB ,B ≤ QR | D | + N x ǫ , where D := X q ,q ∼ Q ( q ,aq )=1( q ,adq )=1 X r ,r ∼ R ( r ,aq r )=1( r ,adq r )=1 q r ≤ B q r ≤ B η d,r ,q η d,r ,q r r q q X n ,n ∼ Nn ≡ n (mod d )( n ,n dr q )=1( n ,n dr q )=1( n q r ,n ) ∈N| n − n |≥ N/ (log x ) C α n α n × X ≤| h |≤ H ′ e (cid:16) ah ( n − n ) dn r q n r q (cid:17) . We wish to show D ≪ N /x ǫ , and so we see it is sufficient to show D ≪ N /x ǫ .We now introduce a new variable s = q r , and then Cauchy in s and q to eliminatethe η d,r ,q coefficients. This gives D ≪ D / (log x ) O (1) QR / , where D := X s ∼ S X q ∼ Q ( q ,ds )=1 (cid:12)(cid:12)(cid:12) X r ∼ R ( r ,ds )=1 X n ,n ∼ N ( n ,n ds )=1( n ,n r q )=1( n q r ,n ) ∈N| n − n |≥ N/ (log x ) C X ≤| h |≤ H ′ ξ r ,q ,n ,n ,h e (cid:16) ah ( n − n ) dn sn q r (cid:17)(cid:12)(cid:12)(cid:12) ≤ X s ψ (cid:16) sS (cid:17) X q ∼ Q ( q ,ds )=1 (cid:12)(cid:12)(cid:12) X r ∼ R ( r ,ds )=1 X n ,n ∼ N ( n ,n ds )=1( n ,n r q )=1( n q r ,n ) ∈N X ≤| h |≤ H ′ ξ r ,q ,n ,n ,h e (cid:16) ah ( n − n ) dn sn r q (cid:17)(cid:12)(cid:12)(cid:12) for some suitable quantity S ∈ [ QR, QR ]and some 1-bounded sequence ξ which does not depend on s (we have absorbedthe conditions on the summation into ξ ). We wish to show D ≪ N /x ǫ , so it issufficient to show that D ≪ N R Q /x ǫ . We expand the square and swap theorder of summation in D to give D ≤ D , RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 67 where D := X q ∼ Q X r ,r ′ ∼ R X n ,n ′ ,n ,n ′ ∼ N ( dn n ′ ,n n ′ r r ′ q )=1 X ≤| h | , | h ′ |≤ H (cid:12)(cid:12)(cid:12) X s ( s,n n ′ q r r ′ )=1 ψ (cid:16) sS (cid:17) e (cid:16) acdn n ′ sn n ′ q r r ′ (cid:17)(cid:12)(cid:12)(cid:12) ,c = h ( n − n ) n ′ r ′ − h ′ ( n ′ − n ′ ) n r . We have the conditions ( n r , n ′ ) = ( n ′ r ′ , n ) = 1 from the construction of N . Wesplit D into two subsums depending on whether c = 0 or not(16.1) D = D c =0 + D c =0 . We first consider D c =0 . We first make a choice of h , n ′ , r ′ and n , for which thereare O ( HN R ) choices. Since ( n , n ) = 1 and c = 0, we see that n | r ′ hn ′ , so thereare at most x o (1) choices of n . We then see that r h ′ ( n ′ − n ′ ) | h ( n − n ) n ′ q ′ ,(and this quantity is non-zero), so there are then at most x o (1) choices of h ′ , n ′ , r .Therefore there are ≪ x o (1) RHN ≪ x ǫ Q R N M choices of n , n , n , n ′ , h, h ′ , r , r ′ with c = 0. Therefore we have D c =0 ≪ QSx ǫ Q R N M ≪ Q R N x − ǫ . (16.2)We now consider D c =0 . We apply Lemma 16.1 to the inner sum, which gives(16.3) X ( s,n n ′ q q q ′ )=1 ψ (cid:16) sS (cid:17) e (cid:16) acdn n ′ sn n ′ q r r ′ (cid:17) ≪ x o (1) Q / N R + x o (1) QRQR N ( c, n n ′ q r r ′ ) . The first term in (16.3) contributes ≪ x o (1) X n ,n ′ ,n ,n ′ ≪ N X q ≪ Q X r ,r ′ ≪ R X | h | , | h ′ |≪ H Q / N R ≪ x o (1) N Q / R H ≪ N Q / R x − ǫ (16.4)to D c =0 . We now consider the second term of (16.3). We note that if e = ( c, n )then since ( n , n n ′ r r ′ ) = 1 we must have that e | h ′ n ′ so ( c, n ) ≤ ( n , hn ′ ). Thus( c, n n ′ n n ′ q r r ′ ) ≤ ( c, q )( c, n )( c, n n ′ n ′ r r ′ ) ≤ ( c, q )( n , hn ′ ) N R . Thus the second term in (16.3) contributes ≪ x o (1) X n ,n ′ ,n ′ ≪ N X r ,r ′ ≪ R X | h | , | h ′ |≪ H N R RN X n ≪ N ( n , hn ′ ) X q ≪ Q ( c, q ) ≪ x o (1) X n ,n ′ ,n ′ ≪ N X r ,r ′ ≪ R X | h | , | h ′ |≪ H N R · x o (1) N · x o (1) Q ≪ x o (1) N QR H to D c =0 , which is smaller that the bound (16.4). Thus(16.5) D c =0 ≪ N Q / R x − ǫ . Putting together (16.1), (16.2) and (16.2), we have that D ≪ Q R N x − ǫ + N Q / R x − ǫ . We wish to show that D ≪ N Q R /x ǫ . The above bound gives this provided Q R < x − ǫ N, (16.6) N Q / R < x − ǫ . (16.7)This gives the result. (cid:3) Proof of Proposition 12.2. First we note that by Lemma 8.7 the set of n, m withmax( | α n | , | β m | ) ≥ (log x ) C has size ≪ x (log x ) O B (1) − C , so by Lemma 8.9 theseterms contribute negligibly if C = C ( A, B ) is large enough. Thus, by dividingthrough by (log x ) C and considering A + 2 C in place of A , it suffices to show theresult when all the sequences are 1-bounded. ( α n still satisfies (6.1) by Lemma13.7.)We factor q = d q and q = d r where P − ( r ) , P − ( q ) > z ≥ P + ( d ) , P + ( d ) intoparts with large and small prime factors. By putting these in dyadic intervals, wesee that it suffices to show for every A > D Q ≍ Q and D R ≍ Q that X d ∼ D P + ( d ) ≤ z X d ∼ D P + ( d ) ≤ z X q ∼ QP − ( q ) >z ( q ,a )=1 X r ∼ RP − ( r ) >z ( r,a )=1 | ∆( qrd d ) | ≪ A x (log x ) A . By Lemma 13.10 we have the result unless D , D ≤ y = x / log log x , and so wemay assume that Q = Q x − o (1) and R = Q x − o (1) . We let d = d d , extendthe summation over d , d to only have the constraint d ≤ y , and then insertcoefficients c q,r,d to remove the absolute values, where we absorb the conditions P − ( qr ) > z ≥ P + ( d ), d ∼ D into the definition of c q,r,d . Thus it suffices to showthat X d ≤ y ( d,a )=1 τ ( d ) X q ∼ Q ( q,a )=1 X r ∼ R ( r,a )=1 c q,r,d ∆( dqr ) ≪ A x (log x ) A . If we let λ b ,b ,b = b =1 X qr = b c q,r,b , then we see that we have a sum of the type considered in Proposition 14.4 (taking‘ R ’ to be QR , ‘ Q ’ to be D and ‘ E ’ to be 1). By the assumptions of the proposition,we have that N QR ≤ N Q Q < x − ǫ , so we have H = ( QDR log x ) /M < E of Proposition 14.4 vanishes. Therefore, by Proposition 14.4, itsuffices to show that E ≪ N Dx ǫ , RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 69 where H = ( DQ R log x ) /M and E := X ( d,a )=1 ψ (cid:16) dD (cid:17) X q ,q ∼ Q ( q ,aq )=1( q ,adq )=1 X r ,r ∼ R ( r ,aq r )=1( r ,adq r )=1 c d,q ,r c d,q ,r dq q r r X n ,n ∼ Nn ≡ n (mod d )( n ,n dq r )=1( n ,n dq r )=1 | n − n |≥ N/ (log x ) C α n α n × X ≤| h |≤ H b ψ (cid:16) hMdq q r r (cid:17) e (cid:16) ahn q r dq r + ahn q r q r (cid:17) . We now see that Lemma 16.2 gives the result provided we have N Q R < x − ǫ ,Q R < x − ǫ N. Recalling that Q = Q x o (1) , and R = Q x − o (1) then gives the result. (cid:3) Zhang-style estimates near x / In this section we prove Proposition 8.2. The argument is a refined version ofthat of Zhang [35]; we take into account cancellation in an additional variable toobtain quantitatively stronger results, but the original work of Zhang (or the similarwork of Polymath [32] would produce qualitatively similar results. These estimatesare key for estimating convolutions where both factors are close to x / ; the mostcritical cases are when one factor is of size Q Q and the other of size x/ ( Q Q ). Lemma 17.1 (Zhang exponential sum estimate) . Let Q, R, M, N satisfy N M ≍ x and Q R < x − ǫ , Q < N < x − ǫ Q , and let H ≪ QN R /x − ǫ . Let c q,r and α n be -bounded complex sequences with c q,r supported on r with P − ( r ) ≥ z . Let Z := X q ∼ Q ( q,a )=1 X r ,r ∼ R ( r ,ar )=1( r ,aqr )=1 X n ,n ∼ Nn ≡ n (mod q )( n ,qr n )=1( n ,qr n )=1 | n − n |≥ N/ (log x ) C α n α n c q,r c q,r qr r × X ≤| h |≤ H b ψ (cid:16) hMqr r (cid:17) e (cid:16) ah (cid:16) n r qr + n qr r (cid:17)(cid:17) , Then we have Z ≪ N Qx ǫ . Proof. We assume throughout that H ≪ QR N/x − ǫ and that Q ≪ N , and deducethe other conditions are sufficient to give the result.Since we only consider r , r with P − ( r r ) ≥ z , r and r have at most (log log x ) prime factors. Therefore, by Lemma 13.2, there are O (exp(log log x ) )) different sets N , N , . . . which cover all possible pairs ( r , r ), and such that if ( r , r ) , ( r ′ , r ′ ) ∈N j then gcd( r , r ′ ) = gcd( r ′ , r ) = 1. Taking the worst such set N , we see that Z ≪ exp((log log x ) ) (cid:12)(cid:12)(cid:12) X q ∼ Q ( q,a )=1 X r ,r ∼ R ( r ,ar )=1( r ,aqr )=1( r ,r ) ∈N X n ,n ∼ Nn ≡ n (mod q )( n ,qr n )=1( n ,qr n )=1 | n − n |≥ N/ (log x ) C α n α n c q,r c q,r qr r × X ≤| h |≤ H b ψ (cid:16) hMqr r (cid:17) e (cid:16) ah (cid:16) n r qr + n qr r (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) . We now Cauchy in n , n and q to eliminate the α -coefficients and insert a smoothmajorant for the n and n summations. This gives (using Q ≪ N ) Z ≪ exp(2(log log x ) ) (cid:16) X q ∼ Q X n ,n ∼ Nn ≡ n (mod q ) q (cid:17) | Z | ≪ x ǫ N Q | Z | , RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 71 where Z := X q ∼ Q ( q,a )=1 X n ,n ( n n ,q )=1 ψ (cid:16) n N (cid:17) ψ (cid:16) n N (cid:17) × (cid:12)(cid:12)(cid:12) X r ,r ∼ R ( r ,ar n )=1( r ,aqr n )=1( r ,r ) ∈N c q,r c q,r r r X ≤| h |≤ H b ψ (cid:16) hMqr r (cid:17) e (cid:16) ah (cid:16) n r qr + n qr r (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) ≤ R | Z | , where Z := X q ∼ Q X r ,r ′ ,r ,r ′ ∼ R ( qr r ′ ,r r ′ )=1 X ≤| h | , | h ′ |≤ H (cid:12)(cid:12)(cid:12) X n ,n n ≡ n (mod q )( n ,qr r ′ )=1( n ,r r ′ )=1 ψ (cid:16) n N (cid:17) ψ (cid:16) n N (cid:17) e (cid:16) c n qr r ′ + c n r r ′ (cid:17)(cid:12)(cid:12)(cid:12) , and where c (mod qr r ′ ) and c (mod r r ′ ) are given by c = a ( hr ′ r ′ − h ′ r r ) r r ′ ,c = a ( hr ′ r ′ − h ′ r r ) qr r ′ . (Here we used the fact that ( r , r ) , ( r ′ , r ′ ) ∈ N to conclude ( r , r ′ ) = ( r ′ , r ) =1.) In order to establish the desired bound Z ≪ N / ( x ǫ Q ), it suffices to show Z ≪ N /x ǫ , and so it suffices to prove(17.1) Z ≪ N R x ǫ . We separate the diagonal terms Z = with hr ′ r ′ = h ′ r r and the off-diagonal terms Z = with hr ′ r ′ = h ′ r r .(17.2) Z ≪ Z = + Z = . We first consider the diagonal terms Z = . Given a choice of h, r ′ , r ′ there are x o (1) choices of h ′ , r , r by the divisor bound. Thus, estimating the remaining sumstrivially we have (using Q ≪ N and H ≪ N QR /x − ǫ )(17.3) Z = ≪ x o (1) QR HN (cid:16) NQ + 1 (cid:17) ≪ N QR x − ǫ . Now we consider the off-diagonal terms Z = . By Lemma 13.5, we have that X n ≡ n (mod q )( n ,r r ′ )=1 ψ (cid:16) n N (cid:17) e (cid:16) c n r r ′ (cid:17) = Nqr r ′ X | ℓ |≤ x ǫ QR /N b ψ (cid:16) ℓ Nqr r ′ (cid:17) S ( c , ℓ q ; r r ′ ) e (cid:16) ℓ n r r ′ q (cid:17) + O ( x − ) . here S ( m, n ; c ) is the standard Kloosterman sum. By Lemma 13.5 again, we havethat X ( n ,qr r ′ )=1 ψ (cid:16) n N (cid:17) e (cid:16) c n + ℓ r r ′ n r r ′ qr r ′ (cid:17) = Nqr r ′ X | ℓ |≤ x ǫ QR /N b ψ (cid:16) ℓ Nqr r ′ (cid:17) S ( c , ℓ q ; r r ′ ) S ( c , ℓ r r ′ + ℓ r r ′ ; q ) + O ( x − ) . Thus, we see that Z is a sum of Kloosterman sums. By the standard Kloostermansum bound of Lemma 13.3 S ( m, n ; c ) ≪ τ ( c ) c / ( m, n, c ) / ≪ c / o (1) ( m, c ) / ,the inner sum has the bound X n ,n n ≡ n (mod q )( n ,qr r ′ )=1( n ,r r ′ )=1 ψ (cid:16) n N (cid:17) ψ (cid:16) n N (cid:17) e (cid:16) c n qr r ′ + c n r r ′ (cid:17) ≪ x o (1) N Q R X | ℓ |≤ x ǫ QR /N | ℓ |≤ x ǫ QR /N Q / R ( c , r r ′ ) / ( c , r r ′ ) / ( c , q ) / ≪ x ǫ Q / R ( hr ′ r ′ − h ′ r r , qr r ′ r r ′ ) / ≪ x ǫ Q / R ( h, r r ) / ( h ′ , r ′ r ′ ) / ( r ′ r ′ , r r )( hr ′ r ′ − h ′ r r , q ) / . Substituting this into our expression for Z = gives Z = ≪ x ǫ Q / R X r ,r ′ ∼ R X r ,r ′ ∼ R ( r r ′ , r r ′ ) X ≤| h | , | h ′ |≤ Hhr ′ r ′ = h ′ r r ( h, r r )( h ′ , r ′ r ′ ) × X q ∼ Q ( hr ′ r ′ − h ′ r r , q ) / ≪ x ǫ Q / R H ≪ N Q / R x − ǫ . (17.4)Substituting (17.3) and (17.4) into (17.2) then gives Z ≪ N QR x − ǫ + N Q / R x − ǫ . This gives the desired bound (17.1) provided we have N < x − ǫ Q , (17.5) Q R < x − ǫ . (17.6)This gives the result. (cid:3) Lemma 17.2 (Second exponential sum estimate) . Let Q, R, M, N ≤ x O (1) satisfy N M ≍ x and N Q < x − ǫ , N Q / R < x − ǫ , N QR < x − ǫ . RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 73 Let α n , λ q,r be -bounded complex sequences, H = ( QR log x ) /M and f Z := X q ∼ Q ( q,a )=1 X r ,r ∼ R ( r r ,a )=1 λ q,r λ q,r φ ( qr ) qr X n ,n ∼ N ( n ,qr )=1( n ,qr )=1 α n α n X ≤| h |≤ H b ψ (cid:16) hMqr (cid:17) e (cid:16) ahn qr (cid:17) . Then we have f Z ≪ N Qx ǫ . Proof. This is essentially [35, § n , n , q, r variables to eliminate the α coefficients,and insert a smooth majorant for n . This gives | f Z | ≤ N log xQ R X q ∼ Q X r ∼ R X n ∼ N X n ( n ,q )=1 ψ (cid:16) n N (cid:17)(cid:12)(cid:12)(cid:12) X r ∼ R ( r ,ar n )=1 X ≤| h |≤ H c q,r ,r ,h e (cid:16) ahn qr (cid:17)(cid:12)(cid:12)(cid:12) ≤ N log xQ R f Z , where c q,r ,r ,h are some 1-bounded coefficients and where f Z := X q ∼ Q X r ,r ′ ∼ R X ≤| h | , | h ′ |≤ H (cid:12)(cid:12)(cid:12) X n ( n,qr r ′ )=1 ψ (cid:16) nN (cid:17) e (cid:16) an ℓqr r ′ (cid:17)(cid:12)(cid:12)(cid:12) ,ℓ := hr ′ − h ′ r . In order to show f Z ≪ N / ( Qx ǫ ) it is sufficient to show f Z ≪ N QR /x ǫ . Wesplit the sum according to whether ℓ = 0 or not(17.7) f Z = f Z ℓ =0 + f Z ℓ =0 . We first consider f Z ℓ =0 , where hr ′ = h ′ r . Given h , r ′ there are x o (1) choices of h ′ , r . Therefore we find(17.8) f Z ℓ =0 ≪ x o (1) QRH N ≪ x o (1) Q R N x . We now consider f Z ℓ =0 . By Lemma 16.1 we have that(17.9) X n ( n,qr r ′ )=1 ψ (cid:16) nN (cid:17) e (cid:16) an ℓqr r ′ (cid:17) ≪ x o (1) Q / R + x o (1) NQR ( ℓ, qr r ′ ) . The first term of (17.9) contributes a total(17.10) ≪ x o (1) QR H Q / R ≪ x o (1) N Q / R x to f Z ℓ =0 . We now consider the second term of (17.9). We see that ( ℓ, qr r ′ ) ≤ ( ℓ, q )( h, r )( h ′ , r ′ )( r , r ′ ) . Thus these terms contribute ≪ x o (1) NQR X r ,r ′ ∼ R ( r , r ′ ) X ≤| h | , | h ′ |≤ H ( h, r )( h ′ , r ′ ) X q ∼ Q ( ℓ, q ) ≪ x o (1) N H R ≪ x o (1) N Q R x . (17.11) Putting together together (17.7), (17.8), (17.10) and (17.11), we see that f Z = f Z ℓ =0 + f Z ℓ =0 ≪ x o (1) N Q / R x + x o (1) N Q R x + x o (1) N Q R x . We recall that it is sufficient to show that f Z ≪ N QR /x ǫ . Thus we are doneprovided we have N Q / R < x − ǫ , (17.12) N Q < x − ǫ , (17.13) N QR < x − ǫ . (17.14)This gives the result. (cid:3) Finally, we are in a position to prove Proposition 8.2. Proof of Proposition 8.2. First we note that by Lemma 8.7 the set of n, m withmax( | α n | , | β m | ) ≥ (log x ) C has size ≪ x (log x ) O B (1) − C , so by Lemma 8.9 theseterms contribute negligibly if C = C ( A, B ) is large enough. Thus, by dividingthrough by (log x ) C and considering A + 2 C in place of A , it suffices to show theresult when all the sequences are 1-bounded. ( α n still satisfies (6.1) by Lemma13.7.)We factor q = rd where P − ( r ) > z ≥ P + ( d ) into parts with large and smallprime factors. By putting these in dyadic intervals, we see that it suffices to showfor every A > DR ≍ Q that X q ∼ Q ( q ,a )=1 X r ∼ RP − ( d ) >z ( r,a )=1 X d ∼ DP + ( d ) ≤ z | ∆( q rd ) | ≪ A x (log x ) A . By Lemma 13.10 we have the result unless D ≤ y = x / log log x , and so we mayassume that R = Q x − o (1) . We let q = q d , and so it suffices to show that for Q = Q x o (1) X q ∼ Q ( q,a )=1 τ ( q ) X r ∼ RP − ( r ) ≥ z ( r,a )=1 | ∆( qr ) | ≪ A x (log x ) A . We insert coefficients c q,r to remove the absolute values, and absorb the condition P − ( r ) > z into the definition of c q,r . We now see that we have a sum of the typeconsidered in Proposition 14.4 with D = E = 1. Therefore, since N > x ǫ Q we mayapply Proposition 14.4, and it suffices to show that E , E ≪ N Qx ǫ , RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 75 where H = ( QR log x ) /M , H = ( QR log x ) /M and where E := X ( q,a )=1 ψ (cid:16) qQ (cid:17) X r ,r ∼ R ( r r ,a )=1 c q,r c q,r φ ( qr ) qr X n ,n ∼ N ( n ,qr )=1( n ,qr )=1 α n α n X ≤| h |≤ H b ψ (cid:16) hMqr (cid:17) e (cid:16) ahn qr (cid:17) , E := X ( q,a )=1 ψ (cid:16) qQ (cid:17) X r ,r ∼ R ( r ,ar )=1( r ,aqr )=1 c q,r c q,r qr r X n ,n ∼ Nn ≡ n (mod q )( n ,n qr )=1( n ,n qr )=1 | n − n |≥ N/ (log x ) C α n α n × X ≤| h |≤ H b ψ (cid:16) hMqr r (cid:17) e (cid:16) ahn r qr + ahn qr r (cid:17) . Absorbing the ψ ( q/Q ) factors into the coefficients c q,r , we see these are preciselythe sums f Z and Z considered in Lemma 17.1 and Lemma 17.2. Thus, theselemmas give the result provided we have Q R < x − ǫ , Q < N < x − ǫ Q ,N Q < x − ǫ , N Q / R < x − ǫ , N QR < x − ǫ . We see that the first two conditions imply the final three (since we may assume QR ≥ x / − ǫ or else the result follows from the Bombieri-Vinogradov theorem).Recalling that Q = Q x o (1) and R = Q x − o (1) then gives the result. (cid:3) Bombieri–Friedlander–Iwaniec-style estimates near x / In this section we prove Proposition 8.3, which is a refinement of [4, Theorem 4]by Bombieri, Friedlander and Iwaniec. Our argument is similar to the work ofBombieri, Friedlander and Iwaniec, but crucially we use the extra flexibility fromour amplification set-up to reduce the contribution of some diagonal terms in thecritical situation of five factors all of length x / . Lemma 18.1 (Deshouillers–Iwaniec Bound) . Let b m , a n be complex sequences, andlet g be a smooth function with k g ( j ) k ∞ ≪ j . Let r ∈ [ R, R ] , s ∈ [ S, S ] and let θ q = max(0 , − λ ( q )) , where λ ( q ) is the least eigenvalue of the congruencesubgroup Γ ( q ) .We have X m ∼ M b m X n ∼ N a n X ( c,r )=1 g (cid:16) cC (cid:17) S ( mr, n, sc ) ≪ x o (1) (cid:16) r S CRM N (cid:17) θ rs k b m kk a n k (cid:16) S RC + M N + SM C + SN C + M N C R (cid:17) / . Proof. This follows from [9, Theorem 9]. (cid:3) Lemma 18.2 (Kim–Sarnak eigenvalue bound) . Let q ∈ Z > and θ q be as in Lemma18.1. Then θ q ≤ / .Proof. This follows from [29, Appendix, Proposition 2]. (cid:3) Lemma 18.3 (Simpler exponential sum estimate) . Let ≤ N, M, Q with N M ≍ x and N / < x − ǫ , Q < x − ǫ . (18.1) Let α n , be a complex sequence with | α n | ≤ x o (1) . Let H := QN (log x ) /x and let e B := X e ∼ E µ ( e ) X q ( q,a )=1 ψ (cid:16) qQ (cid:17) φ ( qe ) q X n ,n ∼ N ( n n ,qe )=1 α n α n X ≤| h |≤ H b ψ (cid:16) − hMq (cid:17) e (cid:16) ahn q (cid:17) . Then we have e B ≪ N Qx ǫ . Proof. This is essentially just the argument of [3, § ahn q = − ahqn + ahn q (mod 1) . Since h ≪ QN (log x ) /x , this implies that e (cid:16) ahn q (cid:17) = e (cid:16) − ahqn (cid:17) + O (cid:16) log xx (cid:17) . RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 77 The error term above contributes ≪ x o (1) N H / ( xQ ) ≪ N / ( x ǫ Q ) since H = QN/x − o (1) < x − ǫ . Thus we find that e B ≪ x o (1) f B + N Qx ǫ , where (replacing h with − h for convenience) f B := X e ∼ E X ≤| h |≤ H X n ,n ∼ N (cid:12)(cid:12)(cid:12) X q ( q,an n )=1 qφ ( qe ) ψ (cid:16) qQ (cid:17) b ψ (cid:16) hMq (cid:17) e (cid:16) ahqn (cid:17)(cid:12)(cid:12)(cid:12) . Using the identity 1 φ ( qe ) = 1 qφ ( e ) X f | qf ∤ e µ ( f ) φ ( f )and M¨obius inversion to remove the condition ( q, an ) = 1, we see that the innersum is bounded bylog xE X f ≪ x ( f ,n e )=1 X f | an ( f ,n )=1 µ ( f ) φ ( f ) (cid:12)(cid:12)(cid:12) X q ( q,n )=1 f f | q q ψ (cid:16) qQ (cid:17) b ψ (cid:16) hMq (cid:17) e (cid:16) ahqn (cid:17)(cid:12)(cid:12)(cid:12) . We let q = f f q ′ , and note that ∂ j ( ∂q ′ ) j (cid:16) ψ (cid:16) q ′ f f Q (cid:17)c ψ (cid:16) − hMq ′ f f (cid:17)(cid:17) ≪ j ( q ′ ) − j . Thus, using partial summation to remove the ψ ( q/Q ) b ψ ( hM/q ) /q weight, we seethat this is ≪ log xQ E X f ≪ x ( f ,n e )=1 X f | an ( f ,n )=1 µ ( f ) φ ( f ) sup Q ′ ,Q ′′ ≍ Q/f f (cid:12)(cid:12)(cid:12) X Q ′ ≤ q ′ ≤ Q ′′ ( q ′ ,n )=1 e (cid:16) ahf f q ′ n (cid:17)(cid:12)(cid:12)(cid:12) . Finally, Lemma 16.1 shows that the inner sum is ≪ N / o (1) + QN − ( h, n ). Thuswe find that f B ≪ x o (1) Q E X e ∼ E X ≤| h |≤ H X n ,n ∼ N (cid:16) N / + QN ( h, n ) (cid:17) ≪ x o (1) H N / Q + x o (1) H NQ ≪ N Qx ǫ/ (cid:16) N / x − ǫ + Qx − ǫ (cid:17) . It suffices to show that f B ≪ N R/ ( Qx ǫ/ ), and so we are done provided N / < x − ǫ , Q < x − ǫ . (18.2)This gives the result. (cid:3) Lemma 18.4 (First reduction of exponential sum) . Let M ≥ x ǫ , Q ≤ x − ǫ , N ≍ KL , M N ≍ x , H := QN (log x ) /x , ≤ E ≤ N/Q , and let α n = X kℓ = nk ∼ Kℓ ∼ L β k γ ℓ for some 1-bounded coefficients β k , γ ℓ . Define B := X e ∼ E µ ( e ) X q ( q,a )=1 q ψ (cid:16) qQ (cid:17) X n ,n ∼ Nn ≡ n (mod qe )( n ,qn )=1( n ,qn )=1 | n − n |≥ N/ (log x ) C α n α n X ≤| h |≤ H b ψ (cid:16) − M hq (cid:17) e (cid:16) ahn q (cid:17) . Then we have for any A > B ≪ (log x ) C + O (1) N / K / Q sup H ′ ≤ HR ≤ aN/QEE ′ ≤ Eθ ∈ [0 , a ′ | a | B ′ | / + O A (cid:16) N Q (log x ) A (cid:17) , where B ′ = B ′ ( θ, a ′ , R, H ′ , E ′ ) is given by B ′ = X r ′ ∼ R X n ∼ N X k ∼ K ( k,r ′ n )=1 r ′ (cid:12)(cid:12)(cid:12) X E ≤ e ≤ E ′ ( e,kn )=1 µ ( e ) X ℓ ∼ Lℓ ≡ kn (mod r ′ e )( ℓ,n )=1 γ ℓ e ( ℓθ ) X h ∼ H ′ e (cid:16) a ′ hr ′ eℓkn (cid:17)(cid:12)(cid:12)(cid:12) . Proof. This is similar to the initial proof of [4, Theorem 4] or [21], but with a slightlydifferent setup. An essentially identical proof goes through, but for completenesswe have included an explicit proof.We note that if ( n , n ) = 1 then we automatically have ( n n , q ) = 1 from n ≡ n (mod q ), and so we may drop the conditions ( n , q ) = ( n , q ) = 1.First we remove the condition ( q, a ) = 1 by M¨obius inversion. This gives B = X e ∼ E µ ( e ) X d | a µ ( d ) X qd | q q ψ (cid:16) qQ (cid:17) X n ,n ∼ Nn ≡ n (mod qe )( n ,n )=1 | n − n |≥ N/ (log x ) C α n α n X ≤| h |≤ H b ψ (cid:16) − M hq (cid:17) e (cid:16) ahn q (cid:17) . We now simplify the exponential. By Bezout’s identity (Lemma 13.1) we have ahn q = − ahqn + ahn q (mod 1) . Since h ≪ ( Q log x ) /M and N M ≍ x this implies that e (cid:16) ahn q (cid:17) = e (cid:16) − ahqn (cid:17) + O (cid:16) log xx (cid:17) . RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 79 The error term above contributes a total ≪ X n = n ∼ N X q,eeq | n − n H log xQx ≪ N H log O (1) xQx . This is acceptably small since H = QN/x − o (1) < x − ǫ . Thus we may replace theexponential with e ( − ahq/m ).We now change variables. Since n ≡ n (mod qe ) and ( n n , q ) = 1 and | n − n | >N/ (log x ) C , we have n − n = qre for some r satisfying N QE (log x ) C ≤ | r | ≤ NQE . We therefore replace the q -summation with a summation over r . Putting | r | intodyadic ranges, we find that B is given by(18.3) B = X d | a µ ( d ) X R =2 j N/ (10 QE log C x ) ≤ R ≤ N/QE B ( d, R ) + O A (cid:16) N Q log A x (cid:17) , where B = B ( d, R ) is given by B := X e ∼ E µ ( e ) X | r |∼ R X n ,n ∼ Nn ≡ n (mod dre )( n ,n )=1( n − n ) /r> | n − n |≥ N/ (log x ) C reα n α n n − n ψ (cid:16) n − n erQ (cid:17) × X ≤| h |≤ H b ψ (cid:16) − M hren − n (cid:17) e (cid:16) − ah ( n − n ) / ( re ) n (cid:17) . Since ( n , n ) = 1, we see that ( n − n ) / ( re ) ≡ − ren (mod n ) so we can simplifythe argument of the exponential to ahren /n .We now wish to remove some of the dependencies between the variables h, n , e and r, n . We separate them in ψ , b ψ by noting that ∂ j + j + j ∂h j ∂n j ∂ j e (cid:16) ren − n ψ (cid:16) n − n erQ (cid:17) b ψ (cid:16) − hM ren − n (cid:17)(cid:17) ≪ j ,j Q | h | − j | n − n | − j | e | − j ≪ (log x ) Cj Q | h | − j | n | − j | e | − j . Therefore, by partial summation we find that B ≪ (log x ) C +1 Q sup H ′′ ≤ HN ′ ≤ NE ′ ≤ E | B | , where B := X r ∼ R X n ∼ N τ ( n ) (cid:12)(cid:12)(cid:12) X E ≤ e ≤ E ′ µ ( e ) X N ≤ n ≤ N ′ n ≡ n (mod dre )( n ,n )=1( n − n ) /r> | n − n |≥ N/ (log x ) C α n X ≤| h |≤ H ′′ e (cid:16) ahren n (cid:17)(cid:12)(cid:12)(cid:12) . (Here we used the fact that β, γ are 1-bounded so | α n | ≤ τ ( n ) and the symmetryin r and − r to just consider postive r .) Finally, we recall that α n = X kℓ = nk ∼ Kℓ ∼ L β k γ ℓ for some 1-bounded coefficients β k , γ ℓ . Substituting this for α n , putting h in dyadicintervals and using the symmetry between h and − h , we see that for some H ′ ≤ H B ≤ log x X r ∼ R X n ∼ N τ ( n ) X k ∼ K ( k,drn )=1 (cid:12)(cid:12)(cid:12) X E ≤ e ≤ E ′ ( e,kn )=1 µ ( e ) X ℓ ∼ Lℓ ≡ kn (mod dre )( ℓ,n )=1 ℓ ∈I ( n ,r,k ) γ ℓ X h ∼ H ′ e (cid:16) ahreℓkn (cid:17)(cid:12)(cid:12)(cid:12) = log x X r ∼ R X n ∼ N X k ∼ K ( k,dren )=1 c n ,r,k X E ≤ e ≤ E ′ ( e,kn )=1 µ ( e ) X ℓ ∼ Lℓ ≡ kn (mod dre )( ℓ,n )=1 ℓ ∈I ( n ,r,k ) γ ℓ X h ∼ H ′ e (cid:16) ahreℓkn (cid:17) , for some suitable coefficients | c n ,r,k | ≤ τ ( n ) and where I ( n , r, k ) is the intervalin [ L, L ] such that ℓ satisfies( n − kℓ ) /r > , | n − kℓ | ≥ N (log x ) C , N ≤ kℓ ≤ N ′ . (We note that ( n , n ) = 1 implies that ( n n , dre ) = 1 since n ≡ n (mod dre ),so we can insert the conditions ( k, dre ) = 1 and ( e, n ) = 1.) We now remove thedependency between ℓ and n , r, k caused by I by noting that ℓ ∈I ( n ,r,k ) = Z e ( ℓθ ) (cid:16) X j ∈I ( n ,r,k ) e ( − jθ ) (cid:17) dθ = Z e ( ℓθ ) c ′ n ,r,k,θ min( L, | θ | − ) dθ, for some suitable 1-bounded coefficients c ′ n ,r,k,θ . (Here we used the standard boundfor an exponential sum over an interval.) This gives B ≤ log x Z min( L, | θ | − ) X | r |∼ R X n ∼ N τ ( n ) X k ∼ K ( k,dren )=1 | B | dθ where B := X E ≤ e ≤ E ′ ( e,kn )=1 µ ( e ) X ℓ ∼ Lℓ ≡ kn (mod dre )( ℓ,n )=1 γ ℓ e ( ℓθ ) X h ∼ H ′ e (cid:16) ahreℓkn (cid:17) . RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 81 Using the L bound Z min( L, | θ | − ) dθ ≪ log x, we obtain B ≪ (log x ) sup θ X r ∼ R X n ∼ N τ ( n ) X k ∼ K ( k,rn )=1 | B | . Finally, applying Cauchy-Schwarz to the outer variables, and replacing r with r ′ = rd and a with a ′ = a/d we see that B ≪ (log x ) N K sup R ′ ≍ R X r ′ ∼ R ′ X n ∼ N X k ∼ K ( k,rn )=1 r ′ | B | , where B is B with d replaced by 1, a replaced with a ′ and r replaced with r ′ .This gives the result. (cid:3) Lemma 18.5 (Improved BFI exponential sum bound, Part I) . Let B ′ = B ′ ( C, D, K, E, H, N ) be given by B ′ := X c ∼ C X d ∼ D ( c,d )=1 X k ∼ K ( k,c )=1 k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X e ∼ E ( e,ck )=1 X h ∼ H X n ∼ Ndn ≡ c (mod ke )( n,kec )=1 β ( h, e, n ) e (cid:16) ahkednc (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Then, if | β ( h, e, n ) | ≤ is supported on square-free e and H ≤ N ≤ C and C, D, K ≪ x O (1) , we have B ′ ≪ ( CDHKN + CEHK N + DH N + DH KN ) log O (1) x + x o (1) DCE sup S ≤ E ≤ ES ≤ KL ≤ ( CE K log x ) / ( DE S S ) E ( B ′′ = + B ′′6 = ) , where B ′′ = = B ′′ = ( S , S , K, E , E, N, H, C, L ) and B ′′6 = = B ′′6 = ( S , S , K, E , E, N, H, C, L ) are given by B ′′ = := X s ∼ S s ∼ S X k ′ ∼ K/s X e ′ ∼ E /s X e ′ ,e ′ ∼ E/e ′ X n ∼ N ( n,s s k ′ e ′ e ′ e ′ )=1 sup θ ∈ R X h ,h ∼ Hh e ′ = h e ′ × (cid:12)(cid:12)(cid:12) X c ′ ( c ′ ,ne ′ e ′ )=1 X ℓ ∼ L g (cid:16) c ′ C/ ( S S ) (cid:17) e ( ℓθ ) S ( a ( h e ′ − h e ′ ) ne ′ e ′ , ℓ ; c ′ s s ) (cid:12)(cid:12)(cid:12) , B ′′6 = := X s ∼ S s ∼ S X k ′ ∼ K/s X e ′ ∼ E /s X e ′ ,e ′ ∼ E/e ′ X n ,n ∼ Nn ≡ n (mod s s k ′ e ′ )( n n ,s s k ′ e ′ )=1( n ,e ′ )=1=( n ,e ′ ) n = n × sup θ ∈ R X h ,h ∼ Hh e ′ n = h e ′ n (cid:12)(cid:12)(cid:12) X c ′ ( c ′ ,n n e ′ e ′ )=1 X ℓ ∼ L g (cid:16) c ′ C/ ( S S ) (cid:17) e ( ℓθ ) S ( r, ℓ ; c ′ s s ) (cid:12)(cid:12)(cid:12) ,r := a ( h e ′ n − h e ′ n ) n n e ′ e ′ , for some smooth function g ( t ) supported on t ≍ and satisfying k g ( j )0 k ∞ ≪ j foreach j ≥ .Proof. First we insert smooth majorants for the c and d summation using thefunction(18.4) f ( t ) := Z ∞ ψ ( y ) ψ (cid:16) ty (cid:17) dyy . We note that f is supported on [1 / , f ≥ , 2] and k f ( j )0 k ∞ ≪ j j ≥ 1. We then expand out the square, giving B ′ ≤ X c f (cid:16) cC (cid:17) X d ( c,d )=1 f (cid:16) dD (cid:17) X k ∼ K ( k,c )=1 k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X e ∼ E ( e,ck )=1 X h ∼ H X n ∼ Ndn ≡ c (mod ke )( n,c )=1 β ( h, e, n ) e (cid:16) ahkednc (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X k ∼ K k X e ,e ∼ E ( e e ,k )=1 X n ,n ∼ Nn ≡ n (mod k )( n ,ke )=1( n ,ke )=1 X h ,h ∼ H β ( h , e , n ) β ( h , e , n ) × X c ( c,e e n n k )=1 f (cid:16) cC (cid:17) X dd ≡ cn (mod ke ) d ≡ cn (mod ke ) f (cid:16) dD (cid:17) e (cid:16) ak ( h e n − h e n ) n n dc (cid:17) . We consider contribution from ‘diagonal’ terms with h e n = h e n and off-diagonal terms with h e n = h e n separately. We let B ′ denote the diagonalterms and B ′ denote the off-diagonal terms, so we have the bound B ′ ≤ B ′ + B ′ . We first consider the contribution from B ′ , which we bound trivially. We see that B ′ ≪ X k ∼ K k X e ,e ∼ En ,n ∼ Nh ,h ∼ Hh e n = h e n X c f (cid:16) cC (cid:17) X d ≡ cn (mod ke ) f (cid:16) dD (cid:17) ≪ K C X b ≤ EHN τ ( b ) (cid:16) DKE + 1 (cid:17) ≪ ( CDHKN + CEHK N ) log O (1) x. This gives the first two terms of the lemma.We now consider B ′ , the terms with h e n = h e n . Let e = ( e , e ) and e = e e ′ , e = e e ′ for some e ′ , e ′ , e which we may assume to be pairwisecoprime since β ( h, e, n ) is supported on square-free e . We see that the inner sum RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 83 is 0 unless n ≡ n (mod e ). Putting e into dyadic ranges, we see that B ′ ≪ log x sup E ≤ E (cid:12)(cid:12)(cid:12) X k ∼ K k X e ∼ E ( e ,k )=1 X e ′ ,e ′ ∼ E/e ( e ′ e ′ ,e k )=1( e ′ ,e ′ )=1 X n ,n ∼ Nn ≡ n (mod ke )( n n ,ke )=1( n ,e ′ )=( n ,e ′ )=1 × X h ,h ∼ Hh e ′ n = h e ′ n β ( h , e e ′ , n ) β ( h , e e ′ , n ) X c ( c,e e ′ e ′ kn n )=1 f (cid:16) cC (cid:17) × X dd ≡ cn (mod ke e ′ ) d ≡ cn (mod ke e ′ ) f (cid:16) dD (cid:17) e (cid:16) ake ( h e ′ n − h e ′ n ) n n dc (cid:17)(cid:12)(cid:12)(cid:12) . By Lemma 13.5 we have that X dd ≡ cn (mod ke e ′ ) d ≡ cn (mod e ′ ) f (cid:16) dD (cid:17) e (cid:16) ake ( h e ′ n − h e ′ n ) n n dc (cid:17) = b f (0) Dcke e ′ e ′ X b (mod d )( b,d )=1 e (cid:16) rbc (cid:17) + Dcke e ′ e ′ X < | ℓ | Let n = ( n , n ), and write n = n n ′ and n = n n ′ for some ( n ′ , n ′ ) = 1.We put n into a dyadic region, and insert 1-bounded coefficients to remove theabsolute values. Thus we see that B ′′6 = ≤ log x sup N ≤ N B ′′ , where B ′′ := X s ∼ S s ∼ S X k ′ ∼ K/s X e ′ ∼ E /s X e ′ ,e ′ ∼ E/e ′ s X n ∼ N ( n ,s s k ′ e ′ e ′ e ′ )=1 X n ′ ,n ′ ∼ N/n n ′ ≡ n ′ (mod s s k ′ e ′ )( n ′ n ′ ,k ′ e ′ s s )=1( n ′ e ′ ,e ′ n ′ )=1 n = n × X h ,h ∼ Hh e ′ n ′ = h e ′ n ′ ξ X ( c ′ ,r )=1 g (cid:16) c ′ S S C (cid:17) X ℓ ∼ L e ( ℓθ ) S ( mr , ℓ ; c ′ s s ) , where ξ = ξ ( s , s , k ′ , e ′ , n , e ′ , e ′ , n ′ , n ′ , h , h ) ∈ C is bounded by 1, where θ = θ ( n , n ′ , n ′ , e , e ′ , e ′ , k ′ , s , s ) ∈ R , and where m := a ( h e ′ n ′ − h e ′ n ′ ) ,r := n n ′ n ′ e ′ e ′ . Let b m = b m ( e ′ , e ′ , n ′ , n ′ , s , s , k ′ , e , n ) be the sequence given by b m := X h ,h ∼ Hm = a ( h e ′ n ′ − h e ′ n ′ ) ξ. We consider dyadic ranges r ∼ R and m ∼ M separately, so taking the worst rangeand dropping some of the summation constraints for an upper bound, we find B ′′ ≤ x o (1) sup M ≪ HEN/E N R ≪ N E /N E L ′ ≤ L X s ∼ S s ∼ S X k ′ ∼ K/s X e ′ ∼ E /s X e ′ ,e ′ ∼ E/e ′ s X n ∼ N X n ′ ,n ′ ∼ N/n n ′ ≡ n ′ (mod s s k ′ e ′ )( n ′ e ′ ,e ′ n ′ )=1 n = n | B ′′ | , where B ′′ := X m ∼ M b m X ℓ ∼ L e ( ℓθ ) X c ′ ( c ′ ,r )=1 g (cid:16) c ′ S S C (cid:17) S ( mr , ℓ ; c ′ s s ) . By Lemma 18.1 and Lemma 18.2, we have that (noting that c ′ is of size C/ ( S S )) B ′′ ≪ x o (1) (cid:16) CRS S LM (cid:17) / (cid:16) C R + LM + C ( M + L ) S S + C LMRS S (cid:17) / L / k b m k . RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 87 We see that X s ∼ S X s ∼ S X k ′ ∼ K/s X e ′ ∼ E /s X n ∼ N X e ′ ,e ′ ∼ E/ ( e ′ s ) X n ′ ,n ′ ∼ N/n n ′ ≡ n ′ (mod s s k ′ e ′ ) n ′ = n ′ ( e ′ n ′ ,e ′ n ′ )=1 k b m k ≪ N X e ′ ,e ′ ≪ E/E X n ′ ,n ′ ≪ N/N ( e ′ n ′ ,e ′ n ′ )=1 n ′ = n ′ τ ( n ′ − n ′ ) X m ∼ M (cid:12)(cid:12)(cid:12) X h ,h ∼ Ha ( h e ′ n ′ − h e ′ n ′ )= m (cid:12)(cid:12)(cid:12) ≪ x o (1) N X m ∼ M X n ′ ≪ N/N X e ′ ≪ E/E X h ∼ H X h ,e ′ ,n ′ h e ′ n ′ | m + h e ′ n ′ ( e ′ n ′ ,e ′ n ′ )=1 × X h ′ ∼ Hh ′ e ′ n ′ ≡ m (mod e ′ n ′ ) X h ′ ∼ Hh ′ e ′ n ′ = m + h ′ n ′ e ′ ≪ x o (1) N X m ∼ M X n ′ ≪ N/N X e ′ ≪ E/E H · x o (1) · (cid:16) HE N EN (cid:17) · ≪ x o (1) EHM N (cid:16) hN EN (cid:17) . Thus X s ∼ S X s ∼ S X k ′ ∼ K/s X e ′ ∼ E /s X n ∼ N X e ′ ,e ′ ∼ E/ ( e ′ s ) X n ′ ,n ′ ∼ N/n n ′ ≡ n ′ (mod s s k ′ e ′ ) n ′ = n ′ ( e ′ n ′ ,e ′ n ′ )=1 k b m k≪ x o (1) (cid:16) HN EN (cid:17) / (cid:16) E HN ME N (cid:17) / . Recalling that L ≪ x o (1) CE K/ ( DE ), R ≍ E N / ( E N ), M ≪ EHN/ ( E N ), S , S ≥ S S ≪ E K and N , E ≥ 1, our bound simplifies to give X s ∼ S X s ∼ S X k ′ ∼ K/s X e ′ ∼ E /s X n ∼ N X e ′ ,e ′ ∼ E/ ( e ′ s ) X n ′ ,n ′ ∼ N/n n ′ ≡ n ′ (mod s s k ′ e ′ ) n ′ = n ′ ( e ′ n ′ ,e ′ n ′ )=1 r ∼ R | B ′′ |≪ x o (1) (cid:16) HN EN (cid:17) / (cid:16) E HN ME N (cid:17) / × (cid:16) CRS S LM (cid:17) / L / (cid:16) C R + LM + C ( M + L ) S S + C LMRS S (cid:17) / ≪ x o (1) (cid:16) HEN (cid:17) / E HN E (cid:16) DNEH (cid:17) / (cid:16) CE KD (cid:17) / (cid:16) C E N + CE HKND (cid:17) / + x o (1) (cid:16) HEN (cid:17) / E HN E (cid:16) DNEHK (cid:17) / (cid:16) CE KD (cid:17) / 28 JAMES MAYNARD × (cid:16) C EHN + C E KD + C EHKDN (cid:17) / . In the final line above we have observed that the maximum occurs when M and L take their largest values and E , N take their smallest values (even after extractinga factor 1 /E ). Moreover, we have observed that for some terms the maximumoccurs when S S = K (these terms have a (1 + DN/ ( EH )) / factor), whereasfor others it occurs when S S = 1 (these terms have a (1 + DN/ ( EHK )) / factor).Since H ≪ EN by assumption, we see the second term is larger than the finalterm in the final set of parentheses, and the (1 + H/ ( EN )) factors are O (1). Thus,simplifying the terms slightly we find that DE CE B ′′ ≪ x o (1) E HK / N / (cid:16) DNEH (cid:17) / (cid:16) CDN + EHK (cid:17) / + x o (1) C / E / HK / N (cid:16) DNEHK (cid:17) / (cid:16) DHN + CEK (cid:17) / . This gives the result. (cid:3) Lemma 18.7 (Improved BFI exponential sum bound, Part III) . Let B ′′ = be as inLemma 18.5 with H ≪ EN . Then we have that B ′′ = ≪ CE x o (1) DE (cid:16) E H K N (cid:17) / (cid:16) HE (cid:17) / (cid:16) DNEH (cid:17) / (cid:16) CDN + EHK (cid:17) / + CE x o (1) DE (cid:16) CE H K N (cid:17) / (cid:16) HE (cid:17) / (cid:16) DNEHK (cid:17) / (cid:16) DH + CEK (cid:17) / . Proof. First we recall that B ′′ = := X s ∼ S s ∼ S X k ′ ∼ K/s X e ′ ∼ E /s X e ′ ,e ′ ∼ E/e ( e ′ ,e ′ )=1 X n ∼ N × sup θ ∈ R (cid:12)(cid:12)(cid:12) X h ,h ∼ H ξ h ,h X c ′ ( c ′ ,ne ′ e ′ )=1 X ℓ g (cid:16) c ′ C/ ( S S ) (cid:17) e ( ℓθ ) S ( mr, ℓ ; c ′ s s ) (cid:12)(cid:12)(cid:12) , for some 1-bounded coefficients ξ h ,h (also depending on s , s , k ′ , e ′ , e ′ , e ′ , n ) andwhere m = a ( h e ′ − h e ′ ) , r = ne ′ e ′ . Let b ′ m be the sequence b ′ m := X h ,h ∼ Ha ( h e ′ − h ′ e ′ )= m ξ h ,h . Thus we have X h ,h ∼ H ξ h ,h X ( c ′ ,ne ′ e ′ )=1 X ℓ g (cid:16) c ′ C/ ( S S ) (cid:17) e ( ℓθ ) S ( mr, ℓ ; c ′ s s )= X M =2 j M ≪ HE/E X m ∼ M b ′ m X ℓ ∼ L e ( ℓθ ) X ( c ′ ,ne ′ e ′ )=1 g (cid:16) c ′ C/ ( S S ) (cid:17) S ( mr, ℓ ; c ′ s s ) . RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 89 By Lemma 18.1 and Lemma 18.2 we have X m ∼ M b ′ m X ℓ ∼ L e ( ℓθ ) X ( c ′ ,ne ′ e ′ )=1 g (cid:16) c ′ C/ ( S S ) (cid:17) S ( mr, ℓ ; c ′ s s ) ≪ x o (1) L / (cid:16) CRS S LM (cid:17) / (cid:16) C R + LM + C ( M + L ) S S + C LMRS S (cid:17) / k b ′ m k . We have that X s ∼ S X s ∼ S X k ′ ∼ K/s X e ′ ∼ E /s X e ′ ,e ′ ∼ E/e ( e ′ ,e ′ )=1 X n ∼ N k b ′ m k ≪ x o (1) KE N X m ∼ M X h ∼ H X e ′ ≪ E/E X e ′ ≪ E/E ( e ′ ,e ′ )=1 e | m + ah e ′ × X h ∼ Hah e ′ = m + ah e ′ X h ′ ∼ Hah ′ ≡ me ′ (mod e ′ ) X h ′ ∼ Hah ′ e ′ = m + ah ′ e ′ ≪ x o (1) EHKM N (cid:16) E HE (cid:17) . Thus we find that B ′′ = ≪ sup M ≪ EH/E L ≪ x o (1) CE K/ ( DE )1 ≪ S S ≪ E KR ≍ E N/E x o (1) (cid:16) EHKM N (cid:17) / (cid:16) E HE (cid:17) / (cid:16) E KNE (cid:17) / × (cid:16) S S RCM L (cid:17) / ( L / ) (cid:16) RC + M L + C ( M + L ) S S + C M LRS S (cid:17) / . The maximum of the expression above clearly occurs when M and L take theirlargest values. (Note that the factors of M / and L / from the first and fifth termsin parentheses are larger than the possible factor ( M L ) / in the denomiator.)Some terms are maximized when S S = 1 and others when S S = E K . Afterremoving a factor 1 /E , we see that the expression is maximized when E = 1.Thus we find that B ′′ = ≪ x o (1) E (cid:16) E H KN (cid:17) / (cid:16) HE (cid:17) / (cid:16) E KN (cid:17) / (cid:16) CE KD (cid:17) / × (cid:16) DNEH (cid:17) / (cid:16) C E N + CE HKD (cid:17) / + (cid:16) DNEHK (cid:17) / (cid:16) C EH + C E KD + C EHKDN (cid:17) / ! . Again, since H ≪ EN by assumption, we see that the second term in the final setof parentheses is larger than the final term. Thus, we obtain B ′′ = ≪ CE x o (1) DE (cid:16) E H K N (cid:17) / (cid:16) HE (cid:17) / (cid:16) DNEH (cid:17) / (cid:16) CDN + EHK (cid:17) / + CE x o (1) DE (cid:16) CE H K N (cid:17) / (cid:16) HE (cid:17) / (cid:16) DNEHK (cid:17) / (cid:16) DH + CEK (cid:17) / . This gives the result. (cid:3) Putting together Lemma 18.5, 18.6 and 18.7, we obtain the following result. Lemma 18.8 (Improved BFI estimate) . Let B ′ = B ′ ( C, D, K, E, H, N ) be as inLemma 18.5. Then, if | β ( h, e, n ) | ≤ is supported on square-free e and H ≪ EN and C, D, K ≪ x O (1) , we have B ′ ≪ ( CDHKN + CEHK N + DH N + DH KN ) log O (1) x + x ǫ ( EK + EN + HK ) / (cid:16) E H KN (cid:17) / (cid:16) DNEH (cid:17) / × (cid:16) CDN + EHK (cid:17) / + x ǫ ( EK + EN + HK ) / (cid:16) C E H K N (cid:17) / (cid:16) DNEHK (cid:17) / . Proof. Applying Lemmas 18.5, 18.6 and 18.7 in turn gives B ′ ≪ ( CDHKN + CEHK N + DH N + DH KN ) log O (1) x + x ǫ (cid:16) CE H KN (cid:17) / (cid:16) DNEHK (cid:17) / (cid:16) DHN + CEK (cid:17) / + x ǫ (cid:16) E H KN (cid:17) / (cid:16) DNEH (cid:17) / (cid:16) CDN + EHK (cid:17) / + x ǫ (cid:16) E H K N (cid:17) / (cid:16) HE (cid:17) / (cid:16) DNEH (cid:17) / (cid:16) CDN + EHK (cid:17) / + x ǫ (cid:16) CE H K N (cid:17) / (cid:16) HE (cid:17) / (cid:16) DNEHK (cid:17) / (cid:16) DH + CEK (cid:17) / . The first line above matches the first line of the lemma. We note that (cid:16) E H KN (cid:17) / (cid:16) DNEH (cid:17) / (cid:16) CDN + EHK (cid:17) / + (cid:16) E H K N (cid:17) / (cid:16) HE (cid:17) / (cid:16) DNEH (cid:17) / (cid:16) CDN + EHK (cid:17) / = ( EK + EN + HK ) / (cid:16) E H KN (cid:17) / (cid:16) DNEH (cid:17) / (cid:16) CDN + EHK (cid:17) / , RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 91 which gives the second line of the lemma. Similarly (cid:16) CE H KN (cid:17) / (cid:16) DNEHK (cid:17) / (cid:16) DHN + CEK (cid:17) / + (cid:16) CE H K N (cid:17) / (cid:16) HE (cid:17) / (cid:16) DNEHK (cid:17) / (cid:16) DH + CEK (cid:17) / = ( EK + EN + HK ) / (cid:16) C E H K N (cid:17) / (cid:16) DNEHK (cid:17) / + ( EK + EN + HK ) / (cid:16) CDE H KN (cid:17) / (cid:16) DNEHK (cid:17) / . The penultimate line above gives the final line of the bound of the lemma. Since H ≪ EN , we see that the final line above is smaller than the bound from thesecond line of the lemma. Thus the result holds. (cid:3) Lemma 18.9. Let B ′ = B ′ ( C, D, K, E, H, N ) be as in Lemma 18.5. Let x / − ǫ/ K, L satisfy Qx ǫ < KL, (18.5) K < x − ǫ Q , (18.6) Q K L < x − ǫ , (18.7) Q K L < x − ǫ . (18.8) Then there is a choice of E satisfying ≤ E ≤ KL/ ( x ǫ Q ) such that sup N ≪ KLR ≪ KL/ ( EQ ) B ′ ( N, K, R, E, H, L ) ≪ K L x ǫ/ . Proof. By Lemma 18.8, the upper bounds for N , R and our bound H ≪ x o (1) QKL/x for H , we see that B ′ = B ′ ( N, K, R, E, H, L ) satisfies B ′ ≪ x o (1) (cid:16) KLK QKLx KLEQ L + KLE QKLx K L E Q L + K Q K L x L (cid:17) + x o (1) K Q K L x KLEQ L + x ǫ (cid:16) E K L E Q + EL + QKLx K L E Q (cid:17) / (cid:16) E Q K L x KLEQ L (cid:17) / × (cid:16) KLE QKLx (cid:17) / (cid:16) KLKL + E QKLx KLEQ (cid:17) / + x ǫ (cid:16) E K L E Q + EL + QKLx K L E Q (cid:17) / (cid:16) K L E Q K L x K L E Q L (cid:17) / × (cid:16) KLE QKLx KLEQ (cid:17) / . This simplifies to give B ′ ≪ x o (1) (cid:16) K L Ex + K L EQx + K L Q x + K L QEx (cid:17) + x ǫ (cid:16) K L EQ + EL + K L E Qx (cid:17) / (cid:16) E K L Qx (cid:17) / (cid:16) xEQ (cid:17) / (cid:16) K L (cid:17) / + x ǫ (cid:16) K L EQ + EL + K L E Qx (cid:17) / (cid:16) EK L x (cid:17) / (cid:16) xKL (cid:17) / . We recall that x / − ǫ/ < Q , and EQ < x ǫ KL < x so that x > KL , x > QE and K L / ( E Qx ) > K L / ( EQ ). Moreover, the conditions on the proposition implythat L, K < x / − ǫ/ < Q , and so K, L > LK/Q and EL > L K / ( E Qx ). Thus B ′ ≪ x o (1) (cid:16) K L Ex + Q K L x (cid:17) + x ǫ (cid:16) EL (cid:17) / (cid:16) QK L E x (cid:17) / (cid:16) xEQ (cid:17) / (cid:16) K L (cid:17) / + x ǫ (cid:16) EL + K L E Qx (cid:17) / (cid:16) EK L x (cid:17) / (cid:16) xKL (cid:17) / . We wish to show that B ′ ≪ K L x − ǫ/ . Most of the important terms are increasingin E , but we need to choose E > K L/x − ǫ/ to handle the first term. Thereforewe choose E = max (cid:16) K Lx − ǫ , (cid:17) . For this to be valid we require that x ǫ EQ < KL , and so we impose the conditions KL > Qx ǫ , (18.9) K < x − ǫ Q . (18.10)If this holds, we then have B ′ K L ≪ x o (1) (cid:16) x ǫ + Q KLx (cid:17) + x ǫ (cid:16) QK L x (cid:17) / (cid:16) x K LQ (cid:17) / + x ǫ (cid:16) QKL x (cid:17) / (cid:16) xQ (cid:17) / + x ǫ (cid:16) K L x + K L x (cid:17) / (cid:16) xKL (cid:17) / + x ǫ min (cid:16) K L Qx , K L Qx (cid:17) / (cid:16) xKL (cid:17) / ≪ x ǫ/ (cid:16) x ǫ + Q KLx (cid:17) + x ǫ (cid:16) Q K L x + Q K L x (cid:17) / + x ǫ (cid:16) K L x + K L x + K L Q x (cid:17) / . This shows that B ′ ≪ K L x − ǫ/ provided we have Q KL < x − ǫ , (18.11) Q K L < x − ǫ , (18.12) Q K L < x − ǫ , (18.13) K L < x − ǫ , (18.14) K L < x − ǫ , (18.15) K L < Q x − ǫ . (18.16) RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 93 We claim that together (18.9), (18.10), (18.12) and (18.13) imply all the otherconditions, and so these conditions simplify to just the ones given in the proposition.Indeed, if (18.10) and (18.12) hold then since x / − ǫ < Q < x / we have KL < K / ( K L ) / < (cid:16) xQ (cid:17) / (cid:16) x Q (cid:17) / = x / Q / < x / ǫ . Thus, since x / ǫ < min( x / − ǫ , Q / x / − ǫ , x − ǫ /Q ) we seethat (18.11), (18.15) and (18.16) follow from (18.10) and (18.12).Similarly, if (18.12) and (18.13) hold then K L = ( K L ) / ( K L ) / < (cid:16) x Q (cid:17) / (cid:16) x Q (cid:17) / = x / Q / < Q x − ǫ . Thus (18.14) follows from (18.12) and (18.13). Thus all the other conditions doindeed follow from (18.9), (18.10), (18.12) and (18.13), giving the result. (cid:3) Proof of Proposition 8.3. This is similar to [4, Theorem 4], but using our refine-ments Lemma 18.9 and Proposition 14.4 given above and keeping track of some ofthe quantities slightly more carefully.First we note that by Lemma 8.7 the set of k, l, m with max( | η n | , | λ ℓ | , | β m | ) ≥ (log x ) B has size ≪ x (log x ) O B (1) − B , so by Lemma 8.9 these terms contributenegligibly if B = B ( A, B ) is large enough. Thus, by dividing through by (log x ) B and considering A + 3 B in place of A , it suffices to show the result when all thesequences are 1-bounded. ( α n still satisfies (6.1) by Lemma 13.7.)We insert coefficients γ q to remove the absolute values, and see that it suffices toshow X q ∼ Q ( q,a )=1 γ q ∆ B ( q ) ≪ A x (log x ) A . This is a special case of the type of sum considered in Proposition 14.4, where D = R = 1, λ q,d,r = 1 and α n = X kℓ = nk ∼ Kℓ ∼ L η k λ ℓ , which satisfies the Siegel-Walfisz estimate since η k does. Since KL > Q (log x ) C byassumption, we have that N > KL > Q (log x ) C . The conditions of the propositionensure that N ≪ KL = ( KL ) / ( K L ) / ≪ x / − ǫ Q / ≪ x / , (18.17)and so M ≫ x/N ≫ x ǫ . Thus, by Proposition 14.4, it suffices to show that for asuitable choice of E ∈ [1 , KL (log x ) − C /Q ] we have E , E ≪ A N Q (log x ) A , where C = C ( A ) is a constant depending only on A , H = ( Q log x ) /M , and E is given by X e ∼ E µ ( e ) X q ( q,a )=1 ψ (cid:16) qQ (cid:17) φ ( qe ) q X n ,n ∼ N ( n n ,qe )=1 α n α n X ≤| h |≤ H b ψ (cid:16) − hMq (cid:17) e (cid:16) ahn q (cid:17) , and E is given by X e ∼ E µ ( e ) X q ( q,a )=1 ψ (cid:16) qQ (cid:17) q X n ,n ∼ Nn ≡ n (mod qe )( n n ,eq )=1( n ,n )=1 | n − n |≥ N/ (log x ) C α n α n X ≤| h |≤ H b ψ (cid:16) − hMq (cid:17) e (cid:16) ahn q (cid:17) . E is of the form considered in Lemma 18.3. The conditions of the propositionimply that Q < x / and N ≪ x / , and so with (18.17) we may apply Lemma18.3 which shows that E ≪ N Qx ǫ . Thus it suffices to consider E . This is precisely the sum considered in Lemma 18.4,and so by Lemma 18.4 it suffices to show for a suitably large constant C = C ( C )depending only on C that B ′ ≪ C L K (log x ) C , where H ′ ≤ H , R ≤ aN/Q , θ ∈ [0 , a ′ | a and B ′ = B ′ ( θ, a ′ , E ′ , R, H ′ ) is givenby B ′ = X r ∼ R X n ∼ N X k ∼ K ( k,n )=1 r (cid:12)(cid:12)(cid:12) X E ≤ e ≤ E ′ ( e,kn )=1 µ ( e ) X ℓ ∼ Lℓ ≡ kn (mod re )( ℓ,n )=1 γ ℓ e ( ℓθ ) X h ∼ H ′ e (cid:16) a ′ hreℓkn (cid:17)(cid:12)(cid:12)(cid:12) . We see that B ′ is of the form B ′ ( N, K, R, E, H, L ) considered in Lemma 18.5 for asuitable choice of coefficients β ( h, e, n ). Moreover, the conditions of the propositionimply that we can apply Lemma 18.9 to bound this, which then gives the result. (cid:3) RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 95 Smoothing preparations Before we embark on the proof of Proposition 11.1, we establish some basic lemmaswhich allow us to pass from sums over integers in an interval with no prime factorslarger than z to a smooth sum of integers. Lemma 19.1 (Smooth partition of unity) . Let C ≥ . There exists smooth non-negative functions e ψ , . . . , e ψ J with J ≤ (log x ) C + 2 such that(1) k e ψ ( j ) i k ∞ ≪ C (( j + 1) log x ) jC for each ≤ i ≤ J and each j ≥ .(2) We have that J X j =1 e ψ j ( t ) = , if t ≤ − / (log x ) C ,O (1) , if − / (log x ) C ≤ t ≤ N, , if ≤ t ≤ ,O (1) , if ≤ t ≤ / (log x ) C , , if / (log x ) C ≤ t. Proof. Recall that ψ is a smooth function supported on [1 / , / 2] and equal to 1on [1 , 2] which satisfies k ψ ( j )0 k ∞ ≤ j j ! . Define J := ⌈ (log x ) C + 1 ⌉ and e ψ i ( t ) := ψ (( t − x ) C − i + 2) , i − / x ) C ≤ t ≤ i − x ) C , , i − x ) C ≤ t ≤ i − / x ) C , − ψ (( t − x ) C − i + 1) , i − / x ) C ≤ t ≤ i (log x ) C , , otherwise.Then it is easy to verify that e ψ i ( t ) is smooth, non-negative, satisfies k e ψ ( j ) i k ∞ ≪ (( j + 1) log x ) jC and that P Jj =1 f ψ j satisfies the bounds of the lemma. (cid:3) Lemma 19.2 (Reduction to smoothed sums) . Let N ≥ x ǫ , N M ≍ x and z ≤ z .Let α m , c q be 1-bounded complex sequences.Imagine that for every choice of N ′ , D, A, C > with N ′ D ≍ N and D ≤ y , andevery smooth function f supported on [1 / , / satisfying k f ( j ) k ∞ ≪ j (log x ) jC ,and for every -bounded complex sequence β d we have the estimate X q ∼ Q c q X m ∼ M α m X d ∼ D β d X n ′ f (cid:16) n ′ N ′ (cid:17)(cid:16) mn ′ d ≡ a (mod q ) − ( mn ′ d,q )=1 φ ( q ) (cid:17) ≪ A,C x (log x ) A . Then for any B > and every interval I ⊆ [ N, N ] we have X q ∼ Q c q X m ∼ M α m X n ∈I P − ( n ) >z (cid:16) mn ≡ a (mod q ) − ( mn,q )=1 φ ( q ) (cid:17) ≪ B x (log x ) B . Proof. We first note that by Lemma 10.3 X n ∈I P − ( n ) >z (cid:16) mn ≡ a (mod q ) − ( mn,q )=1 φ ( q ) (cid:17) ≤ X n ∈I (cid:16)X d | n λ + d (cid:17) mn ≡ a (mod q ) − X n ∈I (cid:16)X d | n λ − d (cid:17) ( mn,q )=1 φ ( q )= X d ≤ y λ + d X dn ′ ∈I (cid:16) mdn ′ ≡ a (mod q ) − ( mdn ′ ,q )=1 φ ( q ) (cid:17) + O (cid:16)(cid:12)(cid:12)(cid:12) X d ≤ y ( λ + d − λ − d ) X dn ′ ∈I ( mdn ′ ,q )=1 φ ( q ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:17) , and similarly we obtain a lower bound with the roles of λ + d and λ − d reversed. ByLemma 10.3 (and recalling that M N ≍ x , N ≥ x ǫ ), the final term above contributes ≪ X q ∼ Q φ ( q ) X m ∼ M (cid:12)(cid:12)(cid:12) X d ≤ y ( d,q )=1 ( λ + d − λ − d ) (cid:16) I φ ( q ) dq + O ( τ ( q )) (cid:17)(cid:12)(cid:12)(cid:12) ≪ M N (log x ) log log x + M x o (1) ≪ B x (log x ) B . Thus it suffices to show that X q ∼ Q c q X m ∼ M α m X d ≤ y λ + d X n ′ ∼ N/ddn ′ ∈I (cid:16) mdn ′ ≡ a (mod q ) − ( mdn ′ ,q )=1 φ ( q ) (cid:17) ≪ B x (log x ) B , and similarly with λ − d in place of λ + d . Let I = [ N , N ]. We give the proof for thecase of λ + d ; the other case is entirely analogous. We first apply Lemma 19.1 to seethat X N /d ≤ n ′ ≤ N /dmdn ′ ≡ a (mod q ) X j ≪ (log x ) B X n ′ mdn ′ ≡ a (mod q ) e ψ j (cid:16) n ′ dN (cid:17) + O (cid:16) X n ′ ∈I d mdn ′ ≡ a (mod q ) (cid:17) , where I d = h N d − N d (log x ) B , N d i ∪ h N d , N d + 2 Nd (log x ) B i . The error term above contributes a total ≪ X d ≤ y X n ′ ∈I d X m ∼ M (cid:16) X q | mn ′ d − a (cid:17) ≪ M N (log x ) O (1) (log x ) B ≪ B x (log x ) B , provided B is large enough (as we may assume). We now split the summation over d into one of O (log x ) B +1 subsums of the form d ∈ [(1 + η ) k , (1 + η ) k +1 ) where η := (log x ) − B . Taking the worst subsum and the worst value of j ≪ (log x ) B ,it suffices to show that X q ∼ Q c q X m ∼ M α m X d ∈ [ d ,d (1+ η )) d ≤ y λ + d X n ′ e ψ j (cid:16) n ′ dN (cid:17)(cid:16) mdn ′ ≡ a (mod q ) − ( mdn ′ ,q )=1 φ ( q ) (cid:17) ≪ B x (log x ) B , RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 97 Since k e ψ ′ j k ∞ ≪ (log x ) B , we can replace e ψ j ( n ′ d/N ) with e ψ j ( n ′ d /N ) at the costof an error ≪ X m ∼ M X d ∈ [ d ,d (1+ η )) X n ′ ≍ N (log x ) B (log x ) B (cid:16) X q | mn ′ d − a (cid:17) ≪ x (log x ) O (1) (log x ) B , and so this is also acceptable. Finally, let ˜ λ d be λ + d with support restricted to d ∈ [ d , d (1 + η )] ∩ [1 , y ], let D := d and let N ′ := N/d . We see that it sufficesto show that X q ∼ Q c q X m ∼ M α m X d ∼ D ˜ λ d X n ′ e ψ j (cid:16) n ′ N ′ (cid:17)(cid:16) mdn ′ ≡ a (mod q ) − ( mdn ′ ,q )=1 φ ( q ) (cid:17) ≪ B x (log x ) B . This now follows from the assumption of the lemma. (cid:3) Triple divisor function estimates In this section we prove Proposition 11.1. This is a generalization on work ofshowing d ( n ) is equidistributed in arithmetic progressions to moduli of size greaterthan x / . Such a result was first obtained by Friedlander and Iwaniec [23] (with anappendix by Birch and Bombieri), was improved by Heath-Brown [27] and Fouvry,Kowalski and Michel [19] and Polymath [32]. It is worth noting that our resultshold with moduli as large as x / , which was the state of the art until the recentwork of [19] and [32] (and is still the record for general residue classes).As with all of these previous results, we rely crucially on Deligne’s results [6, 8]on the Weil Conjectures for general varieties (and its generalizations) to estimatecertain exponential sums. We use these results as a black box in the form of resultsfor correlations of hyper-Kloosterman sums as given by [32], based on the tracefunction formalism developed by Fouvry, Kowalski, Michel and others (see [20] formore details). To optimize our results we require a different arrangement of themanipulations to that of previous works, exploiting the fact that in our situationwe have moduli which have a relatively small factor. The critical case for theseestimates is handling convolutions of three smooth sequences of length x / − / and one rough sequence of length x / , although this is not the bottleneck in ourfinal results.We begin with some notation. LetKl ( a ; q ) := 1 q X b ,b ,b ∈ Z /q Z b b b = a e (cid:16) b + b + b q (cid:17) . be the hyper-Kloosterman sum. We let(20.1) F ( h , h , h ; a, q ) = X b ,b ,b ∈ ( Z /q Z ) × b b b = a e (cid:16) h b + h b + h b q (cid:17) be a closely related sum. Lemma 20.1 (Deligne bound) . Let ( a, q ) = 1 and µ ( q ) = 1 . Then we have | Kl ( a ; q ) | ≪ τ ( q ) . Proof. This is proven in [7, § (cid:3) Lemma 20.2 (Correlations of Hyper-Kloosterman sums) . Let s, r , r be squarefreeintegers with ( s, r ) = ( s, r ) = 1 . Let a , a be integers with ( a , r s ) = 1 =( a , r s ) . Then we have X h ∈ Z ( h,sr r )=1 ψ (cid:16) hH (cid:17) Kl ( a h ; r s )Kl ( a h ; r s ) ≪ ( Hr r s ) ǫ (cid:16) H [ r , r ] s + 1 (cid:17) s / [ r , r ] / gcd( a − a , r , r ) / gcd( a r − a r , s ) / . Proof. This is [32, Corollary 6.26]. We have omitted the condition H ≪ ( s [ r , r ]) O (1) there by adding a factor H ǫ in the bound. (cid:3) RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 99 Lemma 20.3 (Properties of F sum) . We have the following:(1) If ( q , q ) = 1 then F ( h , h , h ; a ; q q ) = F ( h , h , h ; aq ; q ) F ( h , h , h ; aq ; q ) . (2) If ( b, q ) = 1 then F ( h , h , h ; a ; q ) = F ( bh , bh , bh ; ab ; q ) . (3) If ( a, q ) = 1 then F ( h , h , h ; a ; q ) = 0 . (4) If ( a, q ) = 1 and gcd( h , h , h , q ) = d then F ( h , h , h ; a ; q ) = φ ( q ) φ ( q/d ) F (cid:16) h d , h d , h d ; a ; qd (cid:17) . (5) If ( a, q ) = 1 and gcd( h h h , q ) = 1 then F ( h , h , h ; a ; q ) = q Kl ( ah h h ; q ) . (6) If ( a, q ) = 1 and gcd( h h h , q ) = 1 and gcd( h , h , h , q ) = 1 and µ ( q ) =0 then F ( h , h , h ; a ; q ) = 0 . (7) If ( a, q ) = 1 and q | h h h and gcd( h , h , h , q ) = 1 and µ ( q ) = 1 then F ( h , h , h ; a ; q ) depends only on ( h , q ) , ( h , q ) , ( h , q ) and q , and satisfies | F ( h , h , h ; a, q ) | ≪ ( h , q )( h , q )( h , q ) q . Proof. Statement (1) follows from the Chinese Remainder Theorem, letting b i = b i, q + b i, q for unique b i, ∈ ( Z /q Z ) × and b i, ∈ ( Z /q Z ) × .Statement (2) follows from making the change of variables b ′ i = b i b in the summa-tion.Statement (3) is immediate since the sum is supported on b b b ∈ ( Z /q Z ) × .Let p be a prime such that p ℓ || gcd( h , h , h , q ), and let q = p j q with ( q , p ) = 1.By parts (1) and (2), we see that F ( h , h , h ; a ; q ) = F ( h , h , h ; aq ; p j ) F ( h , h , h ; ap j ; q )= F ( h , h , h ; aq ; p j ) F (cid:16) h p ℓ , h p ℓ , h p ℓ ; ap j − ℓ ; q (cid:17) ,F (cid:16) h p ℓ , h p ℓ , h p ℓ ; a ; qp ℓ (cid:17) = F (cid:16) h p ℓ , h p ℓ , h p ℓ ; aq ; p j − ℓ (cid:17) F (cid:16) h p ℓ , h p ℓ , h p ℓ ; ap j − ℓ ; q (cid:17) . Thus it suffices to show (4) when q = p j is a prime power. This is trivial if j = ℓ since then all summands are equal to 1. Otherwise let b i = b ′ i + p j − ℓ b ′′ i 00 JAMES MAYNARD for some b ′ i ∈ ( Z /p j − ℓ Z ) × and b ′′ i ∈ Z /p ℓ Z . The summands in F only depend on b ′ , b ′ and b ′ , so F ( h , h , h ; a ; p j ) is given by X b ′ ,b ′ ,b ′ ∈ ( Z /p j − ℓ Z ) × b ′ b ′ b ′ ≡ a (mod p j − ℓ ) e (cid:16) h b ′ + h b ′ + h b ′ p j (cid:17) X b ′′ ,b ′′ ,b ′′ ∈ Z /p ℓ Z Q i =1 ( b ′ i + p j − ℓ b ′′ i ) ≡ a (mod p j ) . By considering b ′′ i (mod p j − ℓ ), then b ′′ i (mod p j − ℓ ) etc in turn, we see that at eachstage the congruence condition reduces to a linear constraint, and so the inner sumis p ℓ . This gives (4).Statement (5) follows from the change of variables b ′ i = b i h i .From statement (1), it suffices to show statement (6) when q = p j is a prime powerwith j ≥ 2. Without loss of generality let p | h , and let b i = b ′ i + p j − b ′′ i . Then thesummands don’t depend on b ′′ so F ( h , h , h ; a ; p j ) is given by X b ′ ,b ′ ,b ′ ∈ ( Z /p j − Z ) × b ′ b ′ b ′ ≡ a (mod p j − ) e (cid:16) h b ′ + h b ′ + h b ′ p j (cid:17) X b ′′ ,b ′′ ,b ′′ ∈ Z /p Z Q i =1 ( b ′ i + p j − b ′′ i ) ≡ a (mod p j ) e (cid:16) h b ′′ + h b ′′ p (cid:17) . In the inner sum, the congruence condition simplifies to a linear constraint on b ′′ , b ′′ , b ′′ which always has a unique solution b ′′ . Thus b ′′ and b ′′ are unconstrained,so the sum is zero unless p | h and p | h , which is impossible if gcd( h , h , h , q ) = 1.This gives (6).Finally, by statement (1) it suffices to establish (7) when q = p is a prime. Withoutloss of generality, let p | h . In this case we see that F ( h , h , h ; a ; q ) is given by X b ,b ,b ∈ ( Z /p Z ) × b b b ≡ a e (cid:16) h b + h b p (cid:17) = (cid:16) X b ∈ ( Z /p Z ) × e (cid:16) h b p (cid:17)(cid:17) · (cid:16) X b ∈ ( Z /p Z ) × e (cid:16) h b p (cid:17)(cid:17) . The right hand side is a product of Ramanujan sums. These only depend on whether p | h i or not, and the right hand side is bounded by ( h , p )( h , p ) = ( h , p )( h , p )( h , p ) /p .This gives (7). (cid:3) Lemma 20.4 (Reduction to an exponential sum) . Let M N N N ≍ x with x ǫ ≤ N ≤ N ≤ N , and let Q ≥ x ǫ . Let d ∈ [ D, D ] with ( d, a ) = 1 and let ψ , ψ , ψ be smooth functions satisfying k ψ ( j )1 k ∞ , k ψ ( j )2 k ∞ , k ψ ( j )3 k ∞ ≪ j (log x ) Cj and let K = K ( d ) be defined by K := X q ∼ Q ( q,ad )=1 c q X m ∼ M α m X n ,n ,n ψ (cid:16) n N (cid:17) ψ (cid:16) n N (cid:17) ψ (cid:16) n N (cid:17) × (cid:16) mn n n ≡ a (mod qd ) − ( mn n n ,qd )=1 φ ( qd ) (cid:17) , RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 101 where c q , α m are 1-bounded complex sequences (possibly depending on d ). Then wehave for any A > K ≪ A,C xD log A x + N N N D (log x ) O C (1) Q sup H ≤ x ǫ QD/N H ≤ x ǫ QD/N H ≤ x ǫ QD/N b q ,b m ,b ,b ,b (mod d )( b q b m ,d )=1 | K ′ | , where K ′ : = X q ∼ Qq ≡ b q (mod d )( q,a )=1 Q c q q X m ∼ M ( m,q )=1 m ≡ b m (mod d ) α m X ≤| h |≤ H ≤| h |≤ H ≤| h |≤ H h i ≡ b i (mod d ) F ( h , h , h ; amd ; q ) , and where F is the function defined in (20.1) .Proof. This essentially follows from completing the sums in n , n , n . We wish toestimate K = K − K , where K := X q ∼ Q c q φ ( qd ) X m ∼ M α m X n ,n ,n ( mn n n ,qd )=1 ψ (cid:16) n N (cid:17) ψ (cid:16) n N (cid:17) ψ (cid:16) n N (cid:17) , K := X q ∼ Q c q X m ∼ M α m X n ,n ,n mn n n ≡ a (mod qd ) ψ (cid:16) n N (cid:17) ψ (cid:16) n N (cid:17) ψ (cid:16) n N (cid:17) . First we consider K . By Lemma 13.6, we have X ( n,qd )=1 ψ (cid:16) nN (cid:17) = N b ψ (0) φ ( qd ) qd + O C ( x o (1) τ ( qd )) . Therefore, since N ≤ N ≤ N , we have X n ,n ,n ( n n n ,qd )=1 ψ (cid:16) n N (cid:17) ψ (cid:16) n N (cid:17) ψ (cid:16) n N (cid:17) = N N N φ ( qd ) q d b ψ (0) b ψ (0) b ψ (0)+ O C ( N N x o (1) ) . Thus K = K MT + O C (cid:16) x o (1) DN (cid:17) , where K MT := N N N b ψ (0) b ψ (0) b ψ (0) X q ∼ Q ( q,ad )=1 φ ( qd ) c q q d X m ∼ M ( m,qd )=1 α m . Now we consider K . We find by Lemma 13.4 that for H := QDx ǫ /N X n ≡ ab (mod qd ) ψ (cid:16) n N (cid:17) = N qd b ψ (0)+ N qd X ≤| h |≤ H b ψ (cid:16) h N qd (cid:17) e (cid:16) ah bqd (cid:17) + O C ( x − ) . 02 JAMES MAYNARD We first apply this with b = mn n to estimate the n sum in K . By Lemma 13.6,the contribution from the last term above to K is negligible. The contributionfrom the first term to K is X q ∼ Q ( q,ad )=1 N b ψ (0) c q qd X m ∼ M ( m,qd )=1 α m X n ,n ( n n ,qd )=1 ψ (cid:16) n N (cid:17) ψ (cid:16) n N (cid:17) = K MT + O C (cid:16) x o (1) DN (cid:17) . Thus we are left to consider the contribution from the middle term sum over 1 ≤| h | ≤ H . Lemma 13.5 shows that for H := QDx ǫ /N X ( n ,qd )=1 ψ (cid:16) n N (cid:17) e (cid:16) bn qd (cid:17) = N b ψ (0) qd X ( b ,qd )=1 e (cid:16) bb qd (cid:17) + Nqd X ≤| h |≪ H b ψ (cid:16) h N qd (cid:17) X b (mod qd )( b ,qd )=1 e (cid:16) bb + h b qd (cid:17) + O C ( x − ) . The first term above is a multiple of a Ramanujan sum, and so ≪ N gcd( b, qd ) /qd .We apply this with b = ah mn to evaluate the n sum, and then similarly toevaluate the n sum. This gives with H := QDx ǫ /N X n ,n ( n n ,qd )=1 ψ (cid:16) n N (cid:17) ψ (cid:16) n N (cid:17) e (cid:16) ah mn n qd (cid:17) = N N q d X ≤| h |≤ H ≤| h |≤ H b ψ (cid:16) h N qd (cid:17) b ψ (cid:16) h N qd (cid:17) X b ,b ∈ ( Z /qd Z ) × e (cid:16) ab b m + h b + h b qd (cid:17) + O C (cid:16) N N QD ( h , qd ) (cid:17) . Putting this all together, we obtain K = K MT + N N N Q d K + O C (cid:16) x o (1) DN (cid:17) , where K : = X q ∼ Q ( q,ad )=1 Q c q q X m ∼ M ( m,qd )=1 α m X ≤| h |≤ H ≤| h |≤ H ≤| h |≤ H b ψ (cid:16) N h qd (cid:17) b ψ (cid:16) N h qd (cid:17) b ψ (cid:16) N h qd (cid:17) × X b ,b ,b ∈ ( Z /qd Z ) × b b b ≡ am (mod qd ) e (cid:16) h b + h b + h b qd (cid:17) . We see that the final sum over b , b , b is F ( h , h , h ; am ; qd ). We note that for ψ ∈ { ψ , ψ , ψ } we have ∂ j + j ∂q j ∂h j b ψ (cid:16) hNqd (cid:17) ≪ j ,j (log x ) C ( j + j ) q j h j , RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 103 and so we may remove the b ψ factors from K ′ by partial summation. This gives K ≪ C (log x ) O C (1) sup Q ′ ≤ QH ′ ≤ x ǫ QD/N H ′ ≤ x ǫ QD/N H ′ ≤ x ǫ QD/N | K | , where K := X Q ≤ q ≤ Q ′ ( q,ad )=1 Q c q q X m ∼ M ( m,qd )=1 α m X ≤| h |≤ H ′ ≤| h |≤ H ′ ≤| h |≤ H ′ F ( h , h , h ; am ; qd ) . Finally, by Lemma 20.3 and ( q, d ) = 1 we have that F ( h , h , h ; am ; qd ) = F ( h , h , h ; amd ; q ) F ( h , h , h ; amq ; d ) . We note that F ( h , h , h ; amq ; d ) depends only on the values of h , h , h , m, q (mod d )and is trivially bounded by d . Therefore, putting these variables into residueclasses (mod d ) and taking the worst residue classes gives the result. (cid:3) Lemma 20.5 (Dealing with common factors) . Let K ′ be given by K ′ := X Q ≤ q ≤ Q ′ c ′ q X m ∼ M ( m,q )=1 α ′ m X ≤| h |≤ H ≤| h |≤ H ≤| h |≤ H h i ≡ b i (mod d ) ∀ i F ( h , h , h ; amd ; q ) , where c ′ q and α ′ m are 1-bounded complex sequences with c ′ q supported on square-freenumbers coprime to ad . Let K ′ = K ′ ( d , d , e , e , e , Q ′′ , Q ′′′ ) be the related sum,given by K ′ := X Q ′′ ≤ q ≤ Q ′′′ ( q,f f )=1 ˜ c q X m ∼ M ( m,q )=1 α ′ m X ≤| h ′ |≤ H / ( d e )1 ≤| h ′ |≤ H / ( d e )1 ≤| h ′ |≤ H / ( d e ) h ′ i ≡ b ′ i (mod d )( h ′ i ,qd /e i )=1 ∀ i Kl (cid:16) f h ′ h ′ h ′ mf ; q (cid:17) , where f = ae e e , f = d d and where Qd d ≤ Q ′′ ≤ Q ′′′ ≤ Q ′ d d , ˜ c q := qd d c ′ qd d Q . If we have the bound K ′ ≪ d e e e Q ′′′ Mx ǫ for every such choice of Q ′′ , Q ′′′ , d , d , e , e , e and b ′ , b ′ , b ′ (mod d ) , then wehave that K ′ ≪ M Q x ǫ . Proof. To handle possible common factors between the h i and q we use Lemma20.3. Let d = gcd( h , h , h , q ). Then q = d q ′ , h i = d h ′ i for some h ′ , h ′ , h ′ , q ′ 04 JAMES MAYNARD with gcd( h ′ , h ′ , h ′ , q ′ ) = 1. Since c q is supported on square-free q , we only need toconsider ( q ′ , d ) = 1, so F ( h , h , h ; amd , q ) = φ ( d ) F ( h ′ , h ′ , h ′ ; amd ; q ′ ) . Now let d = gcd( h ′ h ′ h ′ , q ′ ), and q ′ = d q ′′ where ( q ′′ , d ) = 1 (since we only needconsider q square-free). We see that F ( h ′ , h ′ , h ′ ; amd ; q ′ ) = F ( h ′ , h ′ , h ′ ; amd ; q ′′ ) F ( h ′ , h ′ , h ′ ; amd q ′′ ; d ) . Since ( h h h , q ′′ ) = 1, we have that F ( h ′ , h ′ , h ′ ; amd d ; q ′′ ) = q ′′ Kl ( h ′ h ′ h amd d ; q ′′ ) . Let h ′ i = h ′′ i e i where e i = gcd( h ′ i , d ). Then we see that F ( h ′ , h ′ , h ′ ; amd q ′′ ; d ) = H ( e , e , e ; d )for some function H which depends only on e , e , e and d and is bounded by e e e /d .Thus K ′ is given by K ′ = X ≤ d ≤ H ( d ,ad )=1 X ≤ d ≤ Q/d ( d ,ad )=1 µ ( d d ) X ≤ e ≤ H /d ≤ e ≤ H /d ≤ e ≤ H /d ( e e e ,d )= d ( e ,e ,e )=1 e i | d ∀ i H ( e , e , e ; d ) X Q/d d ≤ q ′′ ≤ Q ′ / ( d d )( q ′′ ,e e e )=1 c ′ d d q ′′ × X m ∼ M ( m,q ′′ d d )=1 α ′ m X ≤| h ′′ |≤ H /d e ≤| h ′′ |≤ H /d e ≤| h ′′ |≤ H /d e d e i h ′′ i ≡ b i (mod d ) ∀ i ( h ′′ i ,d /e i )=1 ∀ i ( h ′′ i ,q ′′ )=1 ∀ i φ ( d ) q ′′ Kl (cid:16) d e e e h ′′ h ′′ h ′′ amd d ; q ′′ (cid:17) . Let H ′ i := H i /d e i . We recall that H ( e , e , e ) ≪ e e e /d . Then we have K ′ ≪ Q X ≤ d ≤ x ≤ d ≤ x d µ ( d d ) d X e ,e ,e e i | d ∀ i e e e d sup Q/d d ≤ Q ′′ ≤ Q ′′′ ≤ Q ′ /d d b ′′ ,b ′′ ,b ′′ (mod d ) | K ′ | , where K ′ = K ′ ( d , d , e , e , e ) is given by K ′ := X Q ′′ ≤ q ′′ ≤ Q ′′′ ( q ′′ ,e e e )=1 ˜ c q ′′ X m ∼ M ( m,q ′′ d d )=1 α ′ m X ≤| h ′′ |≤ H ′ ≤| h ′′ |≤ H ′ ≤| h ′′ |≤ H ′ h ′′ i ≡ b ′′ i (mod d ) ∀ i ( h ′′ i ,d /e i )=1 ∀ i ( h ′′ i ,q ′′ )=1 ∀ i Kl (cid:16) f h ′′ h ′′ h ′′ mf ; q ′′ (cid:17) , where f = ae e e and f = d d and ˜ c q ′′ = d d q ′′ c q ′′ d d / (2 Q ). Therefore, if K ′ ≪ Q ′′′ M d e e e x ǫ ≪ Q M d d e e e x ǫ , (20.2) RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 105 then, since N N N M ≍ x , we find K ′ ≪ M Q x ǫ X d ,d ≪ x d d X e ,e ,e | d ≪ M Q x ǫ . This gives the result. (cid:3) Lemma 20.6 (Bound for the exponential sum) . Let x ǫ ≤ N ≤ N ≤ N and let H , H , H satisfy H i ≪ x ǫ QR/ ( d e i N i ) . Let K ′′ be given by K ′′ := X q ∼ Qr ∼ R ( qr,f f )=1 ˜ c q,r X m ∼ M ( m,qr )=1 α ′ m X ≤| h |≤ H ≤| h |≤ H ≤| h |≤ H h i ≡ b ′ i (mod d )( h i ,qrd /e i )=1 ∀ i Kl (cid:16) f h h h mf ; qr (cid:17) , where ˜ c q,r are 1-bounded coefficients supported on pairs with qr squarefree, and α m is a 1-bounded complex sequence. Let N , Q, R, M satisfy M Q / R x − ǫ < N < x − ǫ Q R M , (20.3) M Q / R < x − ǫ , (20.4) Then we have that K ′′ ≪ Q R M d e e e x ǫ . Proof. We Cauchy in q , h and h and combine h and h into a single variable h = h h . This gives (dropping some coprimality constraints on the outer variables) K ′′ ≪ H H Q (log x ) O (1) K ′′ , where K ′′ := X q ∼ Q ( q,f f )=1 X ≤| h |≤ H H ( h,q )=1 (cid:12)(cid:12)(cid:12) X ≤| h |≤ H h ≡ b ′ (mod d )( h ,qd /e )=1 X r ∼ R ( r,hh f )=1 X m ∼ M ( m,qr )=1 ˜ c q,r α m Kl (cid:16) f hh mf ; qr (cid:17)(cid:12)(cid:12)(cid:12) . In order to establish the result (recalling M N N N ≍ x and H i ≪ x ǫ QR/ ( d e i N i )),we see it is sufficient to show(20.5) K ′′ ≪ QR M x − ǫ d d e e e N . By symmetry it suffices to just consider h > 0. We put h into dyadic regions h ∼ H with H ≪ H H , take the worst range H . We then insert a smooth majorant forthis h summation for an upper bound, and expand out the square. Dropping somesummation constraints on the outer variables for an upper bound, this gives K ′′ ≤ (log x ) sup H ≤ H H X r ,r ∼ R ( r r ,f f )=1 X q ∼ Q ( q,f f )=1 X m ,m ∼ M ( m ,qr )=1( m ,qr )=1 X ≤| h | , | h ′ |≤ H ( h ,qr )=1( h ′ ,qr )=1 | K ′′ | , where K ′′ = K ′′ ( r , r , q, m , m , h , h ′ ) is given by K ′′ := X ( h,qr r )=1 ψ (cid:16) hH (cid:17) Kl ( f hh m f ; qr )Kl ( f hh ′ m f ; qr ) . 06 JAMES MAYNARD We consider separately the terms with r h ′ m = r h m and those with r h ′ m = r h m .For the ‘diagonal terms’ with r h ′ m = r h m , we use the ‘trivial bound’ K ′′ ≪ x o (1) H which follows from Lemma 20.1. Given r , h ′ , m there are x o (1) choices of r , h and m . Thus these terms contribute to K ′′ a total ≪ X q ∼ Q X r ∼ R X ≤| h ′ |≤ H X m ∼ M x o (1) H ≪ x o (1) QRM H H H ≪ x ǫ Q R MN N N d e e e ≪ Q R M x − ǫ d e e e . (20.6)We now consider the terms when r h ′ m = r h m . In this situation we applyLemma 20.2, which gives that(20.7) K ′ ≪ x ǫ Q / Rd + x ǫ H H Q / [ r , r ] / d d , where d = d ( q, r , r , m , m , h , h ′ ) and d = d ( r , r , h ′ m − h m ) are givenby d = gcd( r h ′ m − r h m , q ) ,d = gcd( h ′ m − h m , r , r ) . Thus the contribution of the first term on the right hand side of (20.7) to K ′ is ≪ x ǫ Q / R X r ,r ∼ R X m ,m ∼ M X ≤| h | , | h ′ |≤ H r h ′ m = r h ′ m X q ∼ Q d ≪ x ǫ Q / R X r ,r ∼ R X m ,m ∼ M X ≤| h | , | h ′ |≤ H r h ′ m = r h ′ m τ ( r h ′ m − r h m ) ≪ x ǫ Q / R M H ≪ x ǫ M Q / R N e d . (20.8)The contribution from the last term on the right hand side of (20.7) to K ′ is ≪ x ǫ H H Q / R X ≤| h | , | h ′ |≤ H X m ,m ∼ Mr h ′ m = r h m X r ,r ∼ R ( r , r ) / d X q ∼ Q d ≪ x ǫ H H Q / R R H M Q ≪ Q / R M e e e d x − ǫ N . (20.9)Thus, recalling that d ≥ e , e , e , together (20.6), (20.8) and (20.9) give K ′ ≪ x − ǫ d d QR Me e e N (cid:16) Q / R Mx − ǫ N + Q / R M x − ǫ + Q R M N x − ǫ (cid:17) . RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 107 This is acceptable for (20.5) if we have N < x − ǫ Q R M , (20.10) M Q / R x − ǫ < N , (20.11) M Q / R < x − ǫ . (20.12)This gives the result. (cid:3) Lemma 20.7. Let x ǫ ≤ N ≤ N ≤ N and x ǫ ≤ M and Q , Q ≥ be such that Q Q ≤ x − ǫ , N N N M ≍ x and M Q / Q x − ǫ ≤ N ≤ x − ǫ Q Q M . Let α m be a 1-bounded complex sequence, I j ⊆ [ N j , N j ] an interval and ∆ K ( q ) := X m ∼ M α m X n ∈I n ∈I n ∈I P − ( n n n ) ≥ z (cid:16) mn n n ≡ a (mod q ) − ( mn n n ,q )=1 φ ( q ) (cid:17) . Then for every A > we have X q ∼ Q ( q ,a )=1 X q ∼ Q ( q ,a )=1 (cid:12)(cid:12)(cid:12) ∆ K ( q q ) (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A . Proof. We first simplify the moduli slightly. Let q be the square-full part of q q ,and q ′ be the square-free part. Then q ′ = q ′ q ′ where q ′ = ( q , q ′ ) and q ′ = ( q , q ′ ).Then we see it clearly suffices to show that X q ∼ Q q square-full X q ′ ∼ Q ′ X q ′ ∼ Q ′ ( q q ′ q ′ ,a )=1 q ′ q ′ square-free (cid:12)(cid:12)(cid:12) ∆ K ( q q ′ q ′ ) (cid:12)(cid:12)(cid:12) ≪ A x (log x ) A , for every choice of Q , Q ′ , Q ′ with Q Q ′ Q ′ ≍ Q Q and Q ≫ Q /Q . By Lemma13.9 we have the result unless Q < z , and so Q ′ = Q x − o (1) , Q ′ = Q x − o (1) . Inparticular, restricting to finer than dyadic intervals and inserting coefficients c q ,q ′ ,q ′ to remove the absolute values, it suffices to show that X q ∼ Q X q ∈ [ Q ′ , Q ′ / X q ∈ [ Q ′ , Q ′ / q q ,aq )=1 q q square-free c q ,q ,q ∆ K ( q q q ) ≪ A x (log x ) A for Q = x o (1) , Q ′ = Q x o (1) , Q ′ = Q x o (1) and for some 1-bounded coefficients c q ,q ,q supported on q q squarefree with ( q q , aq ) = ( q , a ) = 1.Now we reduce to a smoothed situation. By three applications of Lemma 19.2, wesee that it is sufficient to show that for every choice of B > Q ≤ z sup q ∼ Q (cid:12)(cid:12)(cid:12) X q ∈ [ Q ′ , Q ′ / X q ∈ [ Q ′ , Q ′ / c q ,q ,q e ∆ K ( q q q ) (cid:12)(cid:12)(cid:12) ≪ B xQ (log x ) B , 08 JAMES MAYNARD where e ∆ K ( q ) := X d ∼ D d ∼ D d ∼ D e λ d ,d ,d X m ∼ M α m X n ,n ,n f (cid:16) n N ′ (cid:17) f (cid:16) n N ′ (cid:17) f (cid:16) n N ′ (cid:17) × (cid:16) md d d n n n ≡ a (mod q ) − ( md d d n n n ,q )=1 φ ( q ) (cid:17) . and ˜ λ d ,d ,d is a 1-bounded sequence and where D i N ′ i ≍ N i for 1 ≤ i ≤ D i ≤ x o (1) , and where f i are smooth functions supported on [1 / , / 2] satisfying f ( j ) i ≪ j (log x ) jC for all 1 ≤ i ≤ j ≥ C = C ( A ) is a constantdepending only on A . If we let α ′ b := X d d d m = bd i ∼ D i m ∼ M α m e λ d ,d ,d , then we see it suffices to establish the result for ∆ ′ K in place of e ∆ K , where ∆ ′ K ( q )is given by X m ∼ M ′ α ′ m X n ,n ,n f (cid:16) n N ′ (cid:17) f (cid:16) n N ′ (cid:17) f (cid:16) n N ′ (cid:17)(cid:16) mn n n ≡ a (mod q ) − ( mn n n ,q )=1 φ ( q ) (cid:17) . We let c q := X q q = qq ∈ [ Q ′ , Q ′ / q ∈ [ Q ′ , Q ′ / c q ,q ,q , which is a 1-bounded sequence supported on q ∼ Q . By applying Lemma 20.4, wehave thatsup q ∼ Q (cid:12)(cid:12)(cid:12) X q ∼ Q ′ X q ∼ Q ′ c q ,q ,q ∆ ′ K ( q q q ) (cid:12)(cid:12)(cid:12) ≪ A,B xQ (log x ) B + N N N x o (1) ( Q ′ Q ′ ) sup q ∼ Q K ′ , where K ′ := X q ∈ [ Q ′ , Q ′ / q ∈ [ Q ′ , Q ′ / ( Q ′ Q ′ ) c ′ q ,q ,q q q X m ∼ M ′ α ′′ m X ≤| h |≤ H ≤| h |≤ H ≤| h |≤ H h i ≡ b i (mod q ) F ( h , h , h ; amq ; q q ) ,H i := x ǫ Q Q ′ Q ′ N i ≤ x ǫ Q ′ Q ′ N i for i ∈ { , , } , and where α ′′ m is α ′ m restricted to a residue class m ≡ b m (mod q ) and c ′ q ,q ,q is c q ,q ,q restricted to a residue class q q ≡ b q (mod q ). By applying Lemma20.5 we then see it suffices to show that for every choice of e , e , e , d , d with d ≥ e , e , e , and every choice of Q ′′ , Q ′′ with Q ′′ ≤ Q ′ and Q ′′ ≤ Q and everychoice of Q ′′ , Q ′′′ ≍ Q ′ Q ′ /d d and H ′ i ≤ x ǫ Q ′ Q ′ / ( N ′ i d e i ) we have X q ′ ∼ Q ′′ q ′ ∼ Q ′′ c ′′ q ,q ′ ,q ′ X m ∼ M ′ ( m,q ′ q ′ )=1 α ′′ m X ≤| h ′ |≤ H ′ ≤| h ′ |≤ H ′ ≤| h ′ |≤ H ′ h i ≡ b i (mod q )( h ′ i ,q q d /e i )=1 ∀ i Kl ( f h ′ h ′ h ′ mf ; q q ) ≪ d ( Q ′ Q ′ ) Me e e x ǫ , RIMES IN ARITHMETIC PROGRESSIONS TO LARGE MODULI I 109 where c ′′ q ,q ′ ,q ′ = Q ′′ ≤ q ′ q ′ ≤ Q ′′′ ( q ′ q ′ ,d d )=1 X a a = d d q ′ q ′ d d c ′ q ,a q ′ ,a q ′ Q ′ Q ′ . This bound now follows from Lemma 20.6 provided we have M ′ Q ′′ / Q ′′ x − ǫ ≤ N ′ ≤ x − ǫ Q ′′ Q ′′ M ′ ,M ′ Q ′′ / Q ′′ < x − ǫ . Since N ′ = N x o (1) and Q ′′ i ≤ Q ′ i ≤ Q i and M ′ = M x o (1) we see that the abovebounds certainly hold provided we have M Q / Q x − ǫ ≤ N ≤ x − ǫ Q Q M ,M Q / Q < x − ǫ . Finally, if the range for N is non-trivial then we see we must have M Q / Q ≤ x − ǫ , and so the first condition implies the second whenever Q Q ≥ x / − ǫ/ . If instead Q Q ≤ x / − ǫ/ then the whole result follows immediately from the Bombieri-Vinogradov theorem. This completes the proof. (cid:3) Proof of Proposition 11.1. First we note that by Lemma 8.7 the set of m with | α m | ≥ (log x ) C has size ≪ x (log x ) O B (1) − C , so by Lemma 8.9 these terms con-tribute negligibly if C = C ( A, B ) is large enough. Thus, by dividing through by(log x ) C and considering A + C in place of A , it suffices to show the result when | α m | ≤ N N N M ≍ x and N ≥ N ≥ N we have N ≫ x / M / . We first apply Lemma 20.7 with the trivial factorization Q = QR , Q = 1. Thisgives the result provided(20.13) M Q / R / x − ǫ < N < x − ǫ Q R M . We now apply Lemma 20.7 with the factorization Q = Q , Q = R , which givesthe result provided(20.14) M Q / R x − ǫ < N < x − ǫ Q R M . 10 JAMES MAYNARD We see that if Q, R satisfy Q R < x and M < x − ǫ / ( QR ) / then M Q / R x − ǫ = x − ǫ M Q R (cid:16) M Q / R x − ǫ (cid:17) < x − ǫ M Q R (cid:16) Q / R / x ǫ (cid:17) < x − ǫ M Q R . Thus the ranges (20.13) and (20.14) overlap, giving the result for the range M Q / R / x − ǫ < N < x − ǫ Q R M . Since M < x − ǫ Q − / R − / , we see that(20.15) M Q / R / x − ǫ < x / − ǫ M / . Since M < x − ǫ Q − / R − / and Q R < x / , we have(20.16) x − ǫ M Q R > x ǫ Q / R / > x / ǫ . Therefore, using (20.15) and (20.16), we see that (20.13) and (20.14) cover therange x / − ǫ M / ≤ N ≤ x / ǫ . This gives the result. (cid:3) We have now completed the proof of Propositions 8.2, 8.3, 11.1, 12.1 and 12.2, andso completed the proof of Theorem 1.1. References [1] E. Assing, V. Bloomer, and J Li. 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