aa r X i v : . [ m a t h . N T ] S e p AVERAGES OF THE M ¨OBIUS FUNCTION ON SHIFTED PRIMES
JARED DUKER LICHTMAN
Abstract.
It is a folklore conjecture that the M¨obius function exhibits cancellation onshifted primes; that is, P p X µ ( p + h ) = o ( π ( X )) as X → ∞ for any fixed shift h >
0. Weprove the conjecture on average for shifts h H , provided log H/ log log X → ∞ . We alsoobtain results for shifts of prime k -tuples, and for higher correlations of M¨obius with vonMangoldt and divisor functions. Our argument combines sieve methods with a refinementof Matom¨aki, Radziwi l l, and Tao’s work on an averaged form of Chowla’s conjecture. Introduction
Let µ : N → {− , , +1 } denote the M¨obius function, defined multiplicatively on primes p by µ ( p ) = − µ ( p k ) = 0 for k >
2. Many central results in number theory may beformulated in terms of averages of the M¨obius function. Notably, the prime number theoremis equivalent to the statement P n X µ ( n ) = o ( X ), and P n X µ ( n ) = O ( X θ ) for all θ > is equivalent to the Riemann hypothesis.Clearly µ ( p ) = − P p X µ ( p ) = − π ( X ), but less is known about the M¨obiusfunction on shifted primes. It is a folklore conjecture that P p X µ ( p + h ) = o ( π ( X )) forany fixed shift h > h = 1 is Problem 5.2 of [16]). We answer an averaged version of this conjecturewith quantitative bounds. Theorem 1.1. If log H/ log X → ∞ as X → ∞ , then X h H (cid:12)(cid:12)(cid:12)(cid:12) X p X µ ( p + h ) (cid:12)(cid:12)(cid:12)(cid:12) = o ( Hπ ( X )) . (1.1) Further if H = X θ for some θ ∈ (0 , , then for all δ > X h H (cid:12)(cid:12)(cid:12)(cid:12) X p X µ ( p + h ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ δ Hπ ( X )(log X ) / − δ . An immediate consequence is that P p Mathematics Subject Classification. .1. Higher correlations. The influential conjectures of Chowla [1] and Hardy–Littlewood[4] assert that for any fixed tuple H = { h , .., h k } of distinct integers, X n X µ ( n + h ) · · · µ ( n + h k ) = o ( X ) , X n X Λ( n + h ) · · · Λ( n + h k ) = S ( H ) X + o ( X ) , for the singular series S ( H ) = Q p (1 − ν p /p )(1 − /p ) k , where ν p = { h , .., h k (mod p ) } . Both conjec-tures remain open for any k > Theorem 1.3. Suppose log H/ log X → ∞ as X → ∞ , and define ψ δ by ψ δ ( X ) = min n log X log H , (log X ) / − δ o . (1.2) Then for any m, k > , and fixed tuple A = { a , .., a k } of disinct integers, we have X h ,..,h m H (cid:12)(cid:12)(cid:12)(cid:12) X n X m Y j =1 µ ( n + h j ) k Y i =1 Λ( n + a i ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ m,δ, A XH m ψ δ ( X ) m . (1.3)It is worth emphasizing particular aspects of this result. First, (1.3) holds for an arbitraryfixed prime k -tuple. We must average over at least m > µ in order to obtaincancellation. Notably, the cancellation becomes quantitatively stronger for larger m , e.g.error savings (log X ) m/ − δ . For the case m = 0 , k = 2, Matom¨aki-Radziwi l l-Tao [10] handledbinary correlations P n X Λ( n )Λ( n + h ) on average with error savings (log X ) A , though inthe much larger regime H > X / ε .In particular, the Chowla conjecture holds on average along the subsequence of primes. Corollary 1.4. Suppose log H/ log X → ∞ as X → ∞ . Then for any m > , X h ,..,h m H (cid:12)(cid:12)(cid:12)(cid:12) X p X µ ( p + h ) · · · µ ( p + h m ) (cid:12)(cid:12)(cid:12)(cid:12) = o m (cid:0) π ( X ) H m (cid:1) . (1.4)Moreover, using Markov’s inequality we may obtain qualitative cancellation for almost allshifts, with arbitrary log factor savings in the exceptional set. Corollary 1.5. Suppose log H/ log X → ∞ as X → ∞ . Then for any A > , X p X µ ( p + h ) · · · µ ( p + h m ) = o m ( π ( X )) , for all except O A ( H m (log X ) − A ) shifts ( h , .., h m ) ∈ [1 , H ] m . These results build on earlier work of Matom¨aki-Radziwi l l-Tao [9], who established anaverage form of Chowla’s conjecture, X h ,..,h m H (cid:12)(cid:12)(cid:12)(cid:12) X n X µ ( n + h ) · · · µ ( n + h m ) (cid:12)(cid:12)(cid:12)(cid:12) = o m ( XH m ) , (1.5)for any H = H ( X ) → ∞ arbitrarily slowly. Whereas, our results require the faster growth H = (log X ) ψ ( X ) with ψ ( X ) → ∞ arbitrarily slowly. .2. Correlations with divisor functions. Consider fixed integers a > k > l > 2. Thewell studied correlation of two divisor functions d k , d l is predicted to satisfy X n X d k ( n + h ) d l ( n ) = C k,l,h · (cid:0) X + o ( X ) (cid:1) (log X ) k − l − , for a certain (explicit) constant C k,l,h > 0. Recently, Matom¨aki-Radziwi l l-Tao [11] haveshown the following averaged result, in the regime H > (log X ) k log k , X h H (cid:12)(cid:12)(cid:12) X n X d k ( n + h ) d l ( n ) − C k,l,h · X (log X ) k − l − (cid:12)(cid:12)(cid:12) = o k ( HX (log X ) k + l − ) . For higher correlations of divisor functions with M¨obius, we obtain Theorem 1.6. For any j > , k , .., k j > , let k = P ji =1 k i and take any fixed tuple A = { a , .., a j } of distinct integers. If log H/ log X → ∞ , then X h H (cid:12)(cid:12)(cid:12) X n X µ ( n + h ) j Y i =1 d k i ( n + a i ) (cid:12)(cid:12)(cid:12) = o k, A (cid:0) HX (log X ) k − j (cid:1) . Again, we emphasize the need to average over the shift h that inputs to M¨obius µ ( n + h ),while a i may be fixed arbitrarily. Remark . For simplicity, the results are stated for the M¨obius function µ , but our resultshold equally for its completely multiplicative counterpart, the Liouville function λ . In fact,the proof strategy is to reduce from µ to λ .The main number-theoretic input is the classical Vinogradov-Korobov zero-free region (cid:26) σ + it : 1 − σ < c max (cid:8) log q, log( | t | + 3) / log log( | t | + 3) / (cid:9) (cid:27) (1.6)for L ( s, χ ), where χ is a Dirichlet character of modulus q (log X ) A in the Siegel-Walfiszrange, see [6, § Beyond M¨obius. We also consider general multiplicative functions f : N → C , whichdo not pretend to be a character f ( n ) ≈ n it χ ( n ) for some χ (mod q ). More precisely, wefollow Granville and Soundararajan [3] and define the pretentious distance D ( f, g ; X ) = (cid:18) X p X − Re( f ( p ) g ( p )) p (cid:19) / , and the related quantity M ( f ; X, Q ) = inf | t | Xχ ( q ) , q Q D (cid:0) f, n n it χ ( n ); X (cid:1) . (1.7)We may apply recent work of Matom¨aki-Radziwi l l-Tao [12] on Fourier uniformity, in orderto more directly obtain (qualitative) cancellation for averages of non-pretentious multiplica-tive functions over shifted primes. heorem 1.8. Given θ ∈ (0 , let H = X θ . Given a multiplicative function f : N → C with | f | . There exists ρ ∈ (0 , ) such that, if M ( f ; X /H − ρ , Q ) → ∞ as X → ∞ foreach fixed Q > , then X h H (cid:12)(cid:12)(cid:12) X p X f ( p + h ) (cid:12)(cid:12)(cid:12) = o θ,ρ (cid:0) Hπ ( X ) (cid:1) . In particular, f = µ does not pretend to be a Dirichlet character, a fact equivalent to theprime number theorem in arithmetic progressions. Indeed, M ( µ ; X, Q ) > inf | t | Xχ ( q ) , q Q X e (log X )2 / ε p X χ ( p ) p it p > (cid:16) − ε (cid:17) log log X + O (1) , where the latter inequality is well-known to follow from the zero-free region (1.6).1.4. Overview of the proof of Theorem 1.1. We now indicate the general form of theproof. We pursue a variation on the approach of Matom¨aki-Radziwi l l-Tao [9]. Namely, wefirst restrict (1.1) to ‘typical’ terms µ ( n ) for n = p + h ∈ S that have prime factors lyingin certain prescribed intervals [ P , Q ] , [ P , Q ]. The terms with n / ∈ S are sparse, and thusmay be shown to contribute negligibly by standard sieve estimates. (For higher correlations,one may also use sieve estimates, along with work of Henriot [5] to handle a general class offunctions with ‘moderate growth’ that are ‘amenable to sieves.’)Once reduced to numbers with ‘typical factorization,’ we decouple the short interval corre-lation between M¨obius and the indicator for the primes, using a Fourier identity and applyingCauchy-Schwarz (Lemma 2.1). This yields a bound of π ( X ) ≪ X/ log X times a Fourier-typeintegral for µ , sup α Z X (cid:12)(cid:12)(cid:12)(cid:12) X x n x + Hn ∈S µ ( n ) e ( nα ) (cid:12)(cid:12)(cid:12)(cid:12) d x . This decoupling step is a gambit. It has the advantage of only needing to consider µ onits own, but loses a factor of log X from the density of the primes. To make this gambitworthwhile, we must recover over a factor of log X savings in the above Fourier integral for µ .However, Matom¨aki-Radziwi l l-Tao [9, Theorem 2.3] bound the above integral with roughly(log X ) savings (though their bound holds for any non-pretentious multiplicative function g .) Therefore we must refine Matom¨aki-Radziwi l l-Tao’s argument in the special case of g = µ to win back over a full factor of log X . We note this task is impossible unless H is largerthan a power of log (this already hints at why we must assume log H/ log log X → ∞ ).We accomplish this task in the ‘key Fourier estimate’ (Theorem 2.2), which bounds theabove integral with (log X ) A savings for any A > S will implicitly depend on A ).As with [9], this bound is proven by reducing to the analogous estimate with the completelymultiplicative λ , and splitting up α ∈ [0 , 1] into major and minor arcs.The main technical innovation here comes from the major arcs (Proposition 5.1), essen-tially saving a factor (log X ) A in the mean values of ‘typical’ Dirichlet polynomials of theform X X n Xn ∈S λ ( n ) χ ( n ) n s or a character χ of modulus q (log X ) A in the Siegel-Walfisz range. This refines the seminalwork of Matom¨aki-Radziwi l l [7], who obtained a fractional power of log savings for thecorresponding mean values. However, Matom¨aki-Radziwi l l’s results apply to the generalsetting of (non-pretentious) multiplicative functions and appeal to Hal´asz’s theorem, whichoffers small savings. By contrast, our specialization to the M¨obius function affords us thefull strength of Vinogradov-Korobov estimates (Lemma 4.5).The Matom¨aki-Radziwi l l method saves roughly a fractional power of P in the Dirichletmean value when Q ≈ H . So in order to recover from our initial gambit, we are promptedto choose P = (log X ) C for some large C > 0. Then by a standard sieve bound thesize of S is morally O ( log P log Q ) = O C ( log log X log H ). This highlights the need for our assumptionlog H/ log log X → ∞ .We remark that the Matom¨aki-Radziwi l l method requires two intervals [ P , Q ] , [ P , Q ](that define S ) in order to handle ‘typical’ Dirichlet polynomials in the regime H = (log X ) ψ ( X ) for ψ ( X ) → ∞ . Note in general [7] the slower H → ∞ the more intervals we require (thoughby a neat short argument [8], only one interval is needed in the regime H = X θ for θ > Notation The M¨obius function is defined multiplicatively from primes p by µ ( p ) = − µ ( p k ) = 0for k > 2. Similarly the Liouville function λ is defined completely multiplicatively by λ ( p ) = − X ≪ Y and X = O ( Y ) both mean | X | CY forsome some absolute constant C , and X ≍ Y means X ≪ Y ≪ X . If x is a parametertending to infinity, X = o ( Y ) means that | X | c ( x ) Y for some quantity c ( x ) that tends tozero as x → ∞ . Let log k X = log k − (log X ) denote the k th-iterated logarithm.Unless otherwise specified, all sums range over the integers, except for sums over thevariable p (or p , p ,..) which are understood to be over the set of primes P . Let e ( x ) := e πix .We use S to denote the indicator of a predicate S , so S = 1 if S is true and S = 0 if S is false. When S is a set, we write S ( n ) = n ∈S as the indicator function of S . Also let S f denote the function n S ( n ) f ( n ).2. Initial reductions We begin with a Fourier-type bound to decouple correlations of arbitrary functions. Lemma 2.1 (Fourier bound) . Given f, g : N → C , let F ( X ) := P n X | f ( n ) | . Then X | h | H (cid:12)(cid:12)(cid:12)(cid:12) X n X f ( n ) g ( n + h ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ F ( X + 2 H ) · sup α Z X (cid:12)(cid:12)(cid:12)(cid:12) X x n x +2 H g ( n ) e ( nα ) (cid:12)(cid:12)(cid:12)(cid:12) d x . (2.1) Proof. First, the lefthand side of (2.1) is X | h | H (cid:12)(cid:12)(cid:12)(cid:12) X n X f ( n ) g ( n + h ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ H − X | h | H (2 H − | h | ) (cid:12)(cid:12)(cid:12)(cid:12) X n X f ( n ) g ( n + h ) (cid:12)(cid:12)(cid:12)(cid:12) =: H − Σ . (2.2) xpanding the square in Σ and letting h = m − n = m ′ − n ′ , we haveΣ = X | h | H (2 H − | h | ) X n,n ′ X f ( n ) f ( n ′ ) g ( n + h ) g ( n ′ + h )= X n,n ′ X X m,m ′ f ( n ) f ( n ′ ) g ( m ) g ( m ′ ) m − n = m ′ − n ′ · (cid:16) Z X x n,m x +2 H d x (cid:17)(cid:16) Z X x ′ n ′ ,m ′ x ′ +2 H d x ′ (cid:17) . Then orthogonality m − n = m ′ − n ′ = R e (( m − n − m ′ + n ′ ) α ) d α givesΣ = Z Z X X x n,m x +2 H f ( n ) g ( m ) e (cid:0) ( m − n ) α (cid:1) d x · Z X X x ′ n ′ ,m ′ x ′ +2 H f ( n ′ ) g ( m ′ ) e (cid:0) ( n ′ − m ′ ) α (cid:1) d x ′ d α = Z (cid:12)(cid:12)(cid:12)(cid:12) Z X X x n,m x +2 H f ( n ) g ( m ) e (cid:0) ( m − n ) α (cid:1) d x (cid:12)(cid:12)(cid:12)(cid:12) d α . Using Cauchy-Schwarz, we bound Σ asΣ Z Z X (cid:12)(cid:12)(cid:12)(cid:12) X x m x +2 H g ( m ) e ( mα ) (cid:12)(cid:12)(cid:12)(cid:12) d x · Z X (cid:12)(cid:12)(cid:12)(cid:12) X y n y +2 H f ( n ) e ( nα ) (cid:12)(cid:12)(cid:12)(cid:12) d y d α ≪ H (cid:18) sup α Z X (cid:12)(cid:12)(cid:12)(cid:12) X x m x +2 H g ( m ) e ( mα ) (cid:12)(cid:12)(cid:12)(cid:12) d x (cid:19) Z Z X (cid:12)(cid:12)(cid:12)(cid:12) X y n y +2 H f ( n ) e ( nα ) (cid:12)(cid:12)(cid:12)(cid:12) d y d α . (2.3)Using R e ( nα ) d α = n =0 again, the second integral in (2.3) is Z Z X (cid:12)(cid:12)(cid:12)(cid:12) X y n y +2 H f ( n ) e ( nα ) (cid:12)(cid:12)(cid:12)(cid:12) d y d α = Z X X y n,n ′ y +2 H f ( n ) f ( n ′ ) Z e (cid:0) ( n − n ′ ) α (cid:1) d α d y = Z X X y n y +2 H | f ( n ) | d y = X n X +2 H | f ( n ) | Z nn − H d y ≪ HF ( X + 2 H ) . Hence plugging the bound (2.3) for H − Σ back into (2.2) gives the result. (cid:3) Next we consider numbers with ‘typical factorization.’For A, δ > 0, define ψ via H = (log X ) ψ ( X ) so that (1.2) becomes ψ δ ( X ) = min { ψ ( X ) , (log X ) / − δ } . Consider the intervals[ P , Q ] = [(log X ) A , (log X ) ψ ( X ) − A ] , (2.4) [ P , Q ] = [exp (cid:0) (log X ) / δ/ (cid:1) , exp (cid:0) (log X ) − δ/ (cid:1) ] , and define the ‘typical factorization’ set S = S ( X, A, δ ) := { n X : ∃ prime factors p , p | n with p j ∈ [ P j , Q j ] } . (2.5)Using the Fourier bound, we shall reduce Theorem 1.1 to the following. heorem 2.2 (Key Fourier estimate for µ ) . Given any A > , δ > , let S = S ( X, A, δ ) as in (2.5) . Then if log H/ log X → ∞ , sup α Z X (cid:12)(cid:12)(cid:12)(cid:12) X x n x + Hn ∈S µ ( n ) e ( nα ) (cid:12)(cid:12)(cid:12)(cid:12) d x ≪ A,δ HX (log X ) A/ . Proof of Theorem 1.1 from Theorem 2.2. By a standard sieve upper bound [2, Theorem 7.1],for each h H , j = 1 , { p X : q ∤ p + h ∀ q ∈ [ P j , Q j ] } ≪ π ( X ) log P j log Q j hϕ ( h ) . (2.6)Thus, recalling the choice of [ P j , Q j ] in (2.4), the terms p + h / ∈ S trivially contribute to (1.1) X h H (cid:12)(cid:12)(cid:12) X p Xp + h / ∈S µ ( p + h ) (cid:12)(cid:12)(cid:12) X j X h H X p Xq ∤ p + h ∀ q ∈ [ P j ,Q j ] ≪ π ( X ) (cid:16) Aψ ( X ) + (log X ) δ − / (cid:17) X h H hϕ ( h ) ≪ A π ( X ) ψ δ ( X ) X h H hϕ ( h ) ≪ A Hπ ( X ) ψ δ ( X ) . (2.7)On the other hand for p + h ∈ S , Lemma 2.1 with f ( n ) = P ( n ), g ( n ) = S µ ( n ) gives X h H (cid:12)(cid:12)(cid:12)(cid:12) X p Xp + h ∈S µ ( p + h ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ π ( X + 2 H ) · sup α Z X (cid:12)(cid:12)(cid:12)(cid:12) X x n x +2 Hn ∈S µ ( n ) e ( nα ) (cid:12)(cid:12)(cid:12)(cid:12) d x ≪ A,δ HX (log X ) A/ , assuming Theorem 2.2. Thus by Cauchy-Schwarz we obtain X h H (cid:12)(cid:12)(cid:12)(cid:12) X p Xp + h ∈S µ ( p + h ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ (cid:18) H X h H (cid:12)(cid:12)(cid:12)(cid:12) X p Xp + h ∈S µ ( p + h ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) / ≪ HX (log X ) A/ / . (2.8)Hence (2.7) and (2.8) with A = 6 give Theorem 1.1. (cid:3) Let W = (log X ) A . Recall Theorem 2.2 asserts that M H ( X ) ≪ XH/W / for M H ( X ) := sup α Z X (cid:12)(cid:12)(cid:12)(cid:12) X x n x + Hn ∈S µ ( n ) e ( nα ) (cid:12)(cid:12)(cid:12)(cid:12) d x . We first note, that, for technical convenience, it suffices to establish M H ( X ) ≪ XH /W / with H := min { H, exp (cid:0) (log X ) / (cid:1) } . Indeed, if H > H then by the triangle inequality M H ( X ) X k ⌈ H/H ⌉ M H ( X + kH ) ≪ X k ⌈ H/H ⌉ ( X + kH ) H W / ≪ HH · XH W / = XHW / as desired. Hence we may assume H exp (cid:0) (log X ) / (cid:1) hereafter. This reduction is notstrictly necessary, but will simplify the argument. For example, in this case Q < P so theintervals [ P j , Q j ] are disjoint.Consider the ‘refined typical factorization’ sets S d = { n/d : d | n ∈ S} for d < P , that is, S d = S d ( X, A, δ ) = { m X/d : ∃ prime factors p , p | m with p j ∈ [ P j , Q j ] } . (2.9) o far we have reduced Theorem 1.1 to Theorem 2.2 for µ . We now reduce further to theanalogous estimate for its completely multiplicative counterpart λ . Proposition 2.3 (Key Fourier estimate for λ ) . Given any A > , δ > , H = (log X ) ψ ( X ) with ψ ( X ) → ∞ and ψ ( X ) (log X ) / . For d W = (log X ) A and S d = S d ( X, A, δ ) as in (2.9) , we have sup α Z X (cid:12)(cid:12)(cid:12)(cid:12) X x nd x + Hn ∈S d λ ( n ) e ( αn ) (cid:12)(cid:12)(cid:12)(cid:12) d x ≪ A,δ HXd / W / . Proof of Theorem 2.2 from Proposition 2.3. By M¨obius inversion, we have µ = λ ∗ h for h = µ ∗ ( µλ ), where ∗ denotes Dirichlet convolution. That is, h ( d ) = µ ( d ) for squarefree d ,and zero otherwise. Thus we may write X x n x + H S ( n ) µ ( n ) e ( nα ) = X d > h ( d ) X x md x + H S ( md ) λ ( m ) e ( mdα ) , and so the triangle inequality gives Z X (cid:12)(cid:12)(cid:12)(cid:12) X x n x + H S ( n ) µ ( n ) e ( nα ) (cid:12)(cid:12)(cid:12)(cid:12) d x X d > | h ( d ) | Z X (cid:12)(cid:12)(cid:12)(cid:12) X x md x + H S ( md ) λ ( m ) e ( mdα ) (cid:12)(cid:12)(cid:12)(cid:12) d x . (2.10)Note, using the trivial bound and swapping the order of summation and integration, thecontribution of d > W to (2.10) is ≪ X W In this section, we establish Proposition 2.3 by the circle method, following the argumentin [9, Proposition 2.4].Take α ∈ [0 , aq ∈ Q with ( a, q ) = 1and 1 q Q for which (cid:12)(cid:12)(cid:12) α − aq (cid:12)(cid:12)(cid:12) qQ . So we may split [0 , 1] into major arcs M and minor arcs m , according to the size of denomi-nator q compared to W , M = [ q W M ( q ) and m = [0 , \ M , here M ( q ) = S ( a,q )=1 { α : | α − a/q | /qQ } . Recall the definitions (2.4), (2.9),[ P , Q ] = [(log X ) A , (log X ) ψ ( X ) − A ] , [ P , Q ] = [exp (cid:0) (log X ) / δ/ (cid:1) , exp (cid:0) (log X ) − δ/ (cid:1) ] , S d ( X, A, δ ) = { m X/d : ∃ p , p | m with p j ∈ [ P j , Q j ] } . We shall obtain Proposition 2.3 from the following results. Proposition 3.1 (Key minor arc estimate) . Given any A > , H = (log X ) ψ ( X ) with ψ ( X ) → ∞ and ψ ( X ) (log X ) / , let d W = (log X ) A and S d = S d ( X, A, as in (2.9) .Then for any completely multiplicative g : N → C with | g | , we have sup α ∈ m Z X (cid:12)(cid:12)(cid:12)(cid:12) X x nd x + Hn ∈S d g ( n ) e ( nα ) (cid:12)(cid:12)(cid:12)(cid:12) d x ≪ A HXd / W / . Proposition 3.2 (Key major arc estimate for λ ) . Given any A > , δ > , H = (log X ) ψ ( X ) with ψ ( X ) → ∞ and ψ ( X ) (log X ) / , let d W = (log X ) A and S d = S d ( X, A, δ ) as in (2.9) . Then we have sup α ∈ M Z X (cid:12)(cid:12)(cid:12)(cid:12) X x nd x + Hn ∈S d λ ( n ) e ( nα ) (cid:12)(cid:12)(cid:12)(cid:12) d x ≪ A,δ HXdW . We remark that the bounds in the minor arc hold for any bounded multiplicative function,whereas in the major arc the specific choice of λ is needed.3.1. Minor arc. In this subsection, we prove Proposition 3.1. Recall for α ∈ m in the minorarc, | α − a/q | < W /qH with q ∈ [ W, H/W ]. It suffices to show I m := Z R θ ( x ) X x nd x + H S d ( n ) g ( n ) e ( nα ) d x ≪ HX (cid:0) log log XdW (cid:1) / ψ ( X ) , (3.1)uniformly for any α ∈ m and measurable θ : [0 , X ] → C with | θ ( x ) | 1. Letting P = { p : P p Q } , by definition each n ∈ S d has a prime factor in P , so we use a variant of theRamar´e identity S d ( n ) = X p ∈P n = mp S (1) d ( mp ) { q ∈ P : q | m } + p ∤ m , (3.2)where S (1) d = { m X/d : ∃ p | m, p ∈ [ P , Q ] } . As g is completely multiplicative, we obtain I m = X p ∈P X m S (1) d ( mp ) g ( m ) g ( p ) e ( mpα ) { q ∈ P : q | m } + p ∤ m Z R θ ( x ) x mpd x + H d x . Next we split P into dyadic intervals [ P, P ]. It suffices to show for each P ∈ [ P , Q ], X p ∈P P p P X m S (1) d ( mp ) g ( m ) g ( p ) e ( mpα ) { q ∈ P : q | m } + p ∤ m Z R θ ( x ) x mpd x + H d x ≪ HX log P (cid:0) log log XdW (cid:1) / , (3.3) ince then (3.1) will follow by (2.4) and the triangle inequality, using X P ≪ P ≪ Q P =2 j P ≪ X log P ≪ j ≪ log Q j ≪ log log Q log P = log ψ ( X ) − A A ≪ A ψ ( X ) . Fix P . We may replace p ∤ m with 1 in (3.3) at a cost of O ( HX/dP ). Indeed, since theintegral is R R θ ( x ) x mpd x + H d x ≪ H , and S (1) d ( mp ) = 0 unless m X/dP , the cost of suchsubstitution is ≪ X p ∈P P p P X m X/dPp | m H ≪ P XdP H = HXdP . Now the left hand side of (3.3) becomes X m ∈S (1) d g ( m ) { q ∈ P : q | m } + 1 X p ∈P P p P g ( p ) e ( mpα ) mpd X Z R θ ( x ) x mpd x + H d x ≪ X m X/dP (cid:12)(cid:12)(cid:12)(cid:12) X p ∈P P p P g ( p ) e ( mpα ) mpd X Z R θ ( x ) x mpd x + H d x (cid:12)(cid:12)(cid:12)(cid:12) ≪ ( X/dP ) / (cid:18) X m X/dP (cid:12)(cid:12)(cid:12)(cid:12) X p ∈P P p P g ( p ) e ( mpα ) mpd X Z R θ ( x ) x mpd x + H d x (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) / , by the trivial bound and Cauchy-Schwarz. Hence for (3.3) it suffices to show X m X/dP (cid:12)(cid:12)(cid:12)(cid:12) X p ∈P P p P g ( p ) e ( mpα ) mpd X Z R θ ( x ) x mpd x + H d x (cid:12)(cid:12)(cid:12)(cid:12) ≪ H P XW log log P (log P ) . (3.4)We expand the left hand side of (3.4) and sum the resulting geometric series on m , X X p ,p ∈P∩ [ P, P ] Z R g ( p ) g ( p ) θ ( x ) θ ( x ) X m X/dp i ∀ i x i mdp i x i + H e (cid:0) m ( p − p ) α (cid:1) d x d x ≪ HX X p ,p P min (cid:18) HdP , k ( p − p ) α k (cid:19) , since for given d, p , p , there are O ( X ) choices for x and O ( H ) subsequent choices for x since x = x ( p /p ) + O ( H ). Note k z k denotes the distance of z ∈ R to the nearest integer.Thus (3.4) reduces to showing X p ,p P min (cid:18) HP , k ( p − p ) α k (cid:19) ≪ HPW log log P (log P ) . (3.5)The difference of primes is p − p ≪ P . Conversely, any integer n ≪ P may be writtenas n = p − p for p , p P in ≪ nϕ ( n ) P (log P ) − ≪ P log log P (log P ) ways by a standard upper ound sieve, see [2, Proposition 6.22]. Hence for (3.5) it suffices to obtain X n ≪ P min (cid:18) Hn , k nα k (cid:19) ≪ HW ( α ∈ m ) . But this follows by the standard ‘Vinogradov lemma’ [6, p.346]. Lemma 3.3. Given H, P > , take α ∈ [0 , with | α − a/q | /q for some ( a, q ) = 1 . Then X n P min (cid:18) Hn , k nα k (cid:19) ≪ Hq + HP + ( P + q ) log q. Observe H/q + H/P + ( P + q ) log q ≪ H/W since q ∈ [ W, H/W ], P ∈ [ P , Q ] =[ W , H/W ]. This completes the proof in the minor arc.3.2. Major arc. In this subsection, we prove the key major arc estimate assuming thefollowing mean value result for the (twisted) Liouville function. Proposition 3.4. Given A > , δ > , let q W = (log X ) A , d < W , χ (mod q ) , h ∈ [ H/W , H ] , and S d = S d ( X, A, δ ) as in (2.9) . Then for all Y ∈ [ X/W , X ] , we have J d,h,q ( Y ; χ ) := Z YY (cid:12)(cid:12)(cid:12)(cid:12) h X x m x + hm ∈S d λ ( m ) χ ( m ) (cid:12)(cid:12)(cid:12)(cid:12) d x ≪ A,δ YW . (3.6) Proof of Proposition 3.2 from Proposition 3.4. To obtain the key major arc estimate we shallprove the stronger bound, I M := sup α ∈ M Z X (cid:12)(cid:12)(cid:12)(cid:12) X x nd x + H S d ( n ) λ ( n ) e ( nα ) (cid:12)(cid:12)(cid:12)(cid:12) d x ≪ HXdW . (3.7)In the major arc recall α = aq + θ with q W and | θ | W qH . By partial summation with a n = >x/d ( n ) S d ( n ) λ ( n ) e ( na/q ), and A ( t ) = P n t a n , we have X x nd x + H S d ( n ) λ ( n ) e ( nα ) = e ( x + Hd θ ) A ( x + Hd ) − e ( xd θ ) A ( xd ) − πiθ Z ( x + H ) /dx/d e ( tθ ) A ( t ) d t ≪ (cid:12)(cid:12)(cid:12)(cid:12) X x nd x + H S d ( n ) λ ( n ) e ( an/q ) (cid:12)(cid:12)(cid:12)(cid:12) + | θ | Z H/d (cid:12)(cid:12)(cid:12)(cid:12) X x/d n x/d + h S d ( n ) λ ( n ) e ( na/q ) (cid:12)(cid:12)(cid:12)(cid:12) d h . Thus taking the maximizing h and integrating over x ∈ [0 , X ], we obtain I M ≪ I H/d + | θ | Hd sup h H/d I h ≪ I H/d + W qd sup h H/d I h , (3.8)where I h := Z X (cid:12)(cid:12)(cid:12)(cid:12) X x/d n x/d + h S d ( n ) λ ( n ) e ( an/q ) (cid:12)(cid:12)(cid:12)(cid:12) d x . (3.9) hen splitting into residues b (mod q ) gives I h X b ( q ) | e ( ab/q ) | Z X (cid:12)(cid:12)(cid:12)(cid:12) X x/d n x/d + hn ≡ b ( q ) S d ( n ) λ ( n ) (cid:12)(cid:12)(cid:12)(cid:12) d x = X b ( q ) Z X (cid:12)(cid:12)(cid:12)(cid:12) X x/d n x/d + hn ≡ b ( q ) S d ( n ) λ ( n ) (cid:12)(cid:12)(cid:12)(cid:12) d x . Now suppose we have the bound I h ≪ qhXW for h ∈ [ qH/W , H/d ] . (3.10)Then, combining with the trivial bound I h hX when h qH/W , (3.8) becomes I M ≪ I H/d + W qd (cid:16) sup qH/W h H/d I h + sup h qH/W hX (cid:17) ≪ qHXdW + W qd (cid:16) qHXdW + qHXW (cid:17) ≪ HXdW , for q W in the major arc. Hence it suffices to show (3.10).Now to bound I h , we extract the gcd. Let c := ( b, q ) so that c | n , and we let b ′ = b/c , q ′ = q/c , h ′ = h/c , m = n/c . Thus since λ is completely multiplicative, we have I h X c | q | λ ( c ) | X ∗ b ′ ( q ′ ) Z X (cid:12)(cid:12)(cid:12)(cid:12) X x/cd m x/cd + h/cm ≡ b ′ ( q ′ ) S d ( cm ) λ ( m ) (cid:12)(cid:12)(cid:12)(cid:12) d x X c | q cd X ∗ b ′ ( q ′ ) Z X/cd (cid:12)(cid:12)(cid:12)(cid:12) X y m y + h ′ m ≡ b ′ ( q ′ ) S cd ( m ) λ ( m ) (cid:12)(cid:12)(cid:12)(cid:12) d y , using the substitution y = x/cd , and noting S d ( cm ) = S cd ( m ) since c q < P . Thenrecalling orthogonality of characters ϕ ( q ′ ) m ≡ b ′ ( q ′ ) = P χ ( q ′ ) χ ( b ′ ) χ ( m ), we obtain I h X c | q cd X ∗ b ′ ( q ′ ) ϕ ( q ′ ) X χ ( q ′ ) | χ ( b ′ ) | Z X/cd (cid:12)(cid:12)(cid:12)(cid:12) X y m y + h ′ S cd ( m ) λ ( m ) χ ( m ) (cid:12)(cid:12)(cid:12)(cid:12) d y X c | q cd X χ ( q ′ ) Z X/cd (cid:12)(cid:12)(cid:12)(cid:12) X y m y + h ′ S cd ( m ) λ ( m ) χ ( m ) (cid:12)(cid:12)(cid:12)(cid:12) d y . (3.11)We may discard the contribution to (3.11) of the integral over y X/dW , since h ′ = h/c and q W imply an admissible cost ≪ X c | q cdϕ ( q ′ ) XdW h ′ hXW X q ′ | q ϕ ( q ′ ) = qhXW . For the remaining y ∈ [ XdW , Xcd ] in (3.11), we split into dyadic intervals so that I h X c | q cd X χ ( q ′ ) X Y =2 jX dW Y Xcd Z YY (cid:12)(cid:12)(cid:12)(cid:12) X y m y + h ′ S cd ( m ) λ ( m ) χ ( m ) (cid:12)(cid:12)(cid:12)(cid:12) d y + O (cid:16) qHXW (cid:17) . (3.12) y assumption, Proposition 3.4 implies J cd,h ′ ,q ′ ( Y ; χ ) ≪ Y /W , so Cauchy-Schwarz gives Z YY (cid:12)(cid:12)(cid:12)(cid:12) X y m y + h ′ S cd ( m ) λ ( m ) χ ( m ) (cid:12)(cid:12)(cid:12)(cid:12) d y h ′ q Y · J cd,h ′ ,q ′ ( Y ; χ ) ≪ A Y h ′ W . So plugging back into (3.12), we obtain I h ≪ X c | q cdϕ ( q ′ ) X Y =2 jX W Y Xcd Y h ′ W = hdW X q ′ | q ϕ ( q ′ ) X Y =2 jX dW We collect some standard lemmas on Dirichlet polynomials.The first is the integral mean value theorem [6, Theorem 9.1]. Lemma 4.1 (mean value) . For D ( s ) = P n N a n n − s , we have Z T − T | D ( it ) | d t = ( T + O ( N )) X n N | a n | . One may discretize the mean value theorem by replacing the intergal over [ − T, T ] with asum over a well-spaced set W ⊂ [ − T, T ]. Definition 4.2. A set W ⊂ R is well-spaced if | w − w ′ | > w, w ′ ∈ W .Next is the Hal´asz-Montgomery inequality [6, Theorem 9.6], which offers an improvementto the (discretized) mean value theorem when the well-spaced set is ‘sparse.’ Lemma 4.3 (Hal´asz-Montgomery) . Given D ( s ) = P n N a n n − s and a well-spaced set W ⊂ [ − T, T ] . Then X t ∈W | D ( it ) | ≪ ( N + |W|√ T ) log 2 T X n N | a n | . We also need a bound on the size of well-spaced sets W in terms of the values of primeDirichlet polynomials on 1 + i W [7, Lemma 8]. Lemma 4.4. Let a p ∈ C be indexed by primes, with | a p | , and define the prime polynomial P ( s ) = X L p L a p p s . Suppose a well-spaced set W ⊂ [ − T, T ] satisfies | P (1 + it ) | > /U for all t ∈ W . Then |W| ≪ U T U +log log T ) / log L . Lemma 4.5. Given A, K > , θ > , and a Dirichlet character χ mod q (log X ) A . Assume exp (cid:0) (log X ) θ (cid:1) P Q X , and let P ( s, χ ) = P P p Q χ ( p ) p − s . Then for any | t | X , | P (1 + it, χ ) | ≪ A,K,θ log X | t | + (log X ) − K . roof. This follows as with [8, Lemma 2], except that the Vinogradov–Korobov zero-freeregion for ζ ( s ) is replaced by that of L ( s, χ ). (cid:3) We also use a Parseval-type bound. This shows that the average of a multiplicativefunction in almost all short intervals can be approximated by its average on a long interval,provided the mean square of the corresponding Dirichlet polynomial is small. Lemma 4.6 (Parseval bound) . Given T ∈ [(log X ) / , X / ] , and take a sequence ( a m ) ∞ m =1 with | a m | . Assume h h X/T . For x ∈ [ X, X ] , define S j ( x ) = X x m x + h j a m , and A ( s ) = X X m X a m m s . Then X Z XX (cid:12)(cid:12)(cid:12) h S ( x ) − h S ( x ) (cid:12)(cid:12)(cid:12) d x ≪ T + Z X/h T | A (1 + it ) | d t + max T > X/h X/h T Z TT | A (1 + it ) | d t . Proof. This follows as in [7, Lemma 14] with (log X ) / replaced by general T . (cid:3) We have a general mean value of products, via the Ramar´e identity [7, Lemma 12]. Lemma 4.7. For V, P, Q > , denote P = [ P, Q ] ∩ P . Let a m , b m , c p be bounded sequences forwhich a mq = b m c q when q ∤ m and q ∈ P . Let Q v,V ( s ) = X q ∈P e v/V q e ( v +1) /V c q q s ,R v,V ( s ) = X Xe − v/V m Xe − v/V b m m s · { p | m : p ∈ P} + 1 , and take a measurable set T ⊂ [ − T, T ] . Then for I = [ ⌊ V log P ⌋ , V log Q ] ∩ Z , we have Z T (cid:12)(cid:12)(cid:12) X X n X a n n it (cid:12)(cid:12)(cid:12) d t ≪ V log (cid:0) QP (cid:1) X v ∈I Z T | Q v,V (1 + it ) R v,V (1 + it ) | d t + (cid:16) TX + 1 (cid:17)(cid:16) V + 1 P + X X n Xp ∤ n ∀ p ∈P | a n | n (cid:17) In the next result we employ the Fundamental Lemma of the sieve, along with the Siegel-Walfisz theorem. Lemma 4.8. Given A, K > , q (log x ) A , Dirichlet character χ (mod q ) , and let D = Q p ∈P p for any set of primes P ⊂ ( q, x / log log x ) . Then X m x ( m, D )=1 λ ( m ) χ ( m ) ≪ A,K x (log x ) K . roof. First partition the sum on m by the values of λ ( m ) , χ ( m ), S := X m x ( m, D )=1 λ ( m ) χ ( m ) = X b ( q ) ,ν ∈{± } νχ ( b ) X m ∈A ( b,ν ) ( m, D )=1 (4.1)for the set A ( b,ν ) = { m x : m ≡ b ( q ) , λ ( m ) = ν } .Now it suffices to prove X m ∈A ( b,ν ) ( m, D )=1 = x q Y p |D (cid:16) − p (cid:17) + O A,K (cid:0) x (log x ) − K (cid:1) (4.2)uniformly in b, ν , from which it will follow S = x qd Y p |D (cid:16) − p (cid:17) X b ( q ) ,ν ∈{± } νχ ( b ) + O (cid:0) x (log x ) − K (cid:1) ≪ x (log x ) K , by pairing up terms ν = ± 1. This will give the lemma.Now to show (4.2), write A = A ( b,ν ) . For d | D the set of multiples A d = { m ∈ A : d | m } has size |A d | = X m xd | m, m ≡ b ( q ) λ ( m )= ν = X n x/dnd ≡ b ( q ) νλ ( nd ) + 12 = νλ ( d )2 X n x/dn ≡ bd − ( q ) λ ( n ) + x qd + O ( q )noting ( q, d ) = 1 = ( q, D ). Moreover max c ( q ) (cid:12)(cid:12) P n yn ≡ c ( q ) λ ( n ) (cid:12)(cid:12) ≪ A,K y (log y ) − K by Siegel-Walfisz, which is valid by the assumption q (log X ) A . Thus |A d | = x qd + R d , where | R d | ≪ xd log( x/d ) − K + q. (4.3)Now for any D > ( m, D )=1 is bounded in between P d | ( m, D ) d In this section we prove Proposition 3.4 based on the following mean value theorem forDirichlet polynomials with typical factorization. This refines Matom¨aki-Radziwi l l [7, Propo-sition 12] in the case of g = λ χ , by leveraging Vinogradov-Korobov type bounds. Proposition 5.1. Given any A > , δ > , denote B = 11 A . Write H = (log X ) ψ ( X ) with ψ ( X ) → ∞ and ψ ( X ) (log X ) / . Take q (log X ) A , d < (log X ) A , a Dirichlet character χ (mod q ) , and let S d = S d ( X, A, δ ) as in (2.9) . For any Y ∈ [ X / , X ] , define G ( s ) = X Y n Yn ∈S d λ ( n ) χ ( n ) n s . Then for any T ∈ [ Y / , Y ] , we have Z T (log X ) B | G (1 + it ) | d t ≪ A,δ (cid:16) Q TY + 1 (cid:17) (log X ) − B . (5.1) Proof of Proposition 3.4 from Proposition 5.1. We shall prove J := Z YY (cid:12)(cid:12)(cid:12) h S h ( x ) (cid:12)(cid:12)(cid:12) d x ≪ A,δ Y (log X ) A (5.2)for Y ∈ [ X/W , X ], h = h ∈ [ qH/W , H ] and the sum S l ( x ) := X x m x + lm ∈S d λ ( m ) χ ( m ) . First we claim S x (0) ≪ x (log x ) − K for all K > 0. To this, recall from (2.9) that each m ∈ S d has prime factors p , p | m with p j ∈ [ P j , Q j ]. So by inclusion-exclusion, theindicator is S d ( m ) = P j =0 ( − j ( m, D j )=1 where D j = Q p ∈P j p for the sets of primes P = ∅ , P = [ P , Q ], P = [ P , Q ], P = P ∪ P . Hence applying Lemma 4.8 to each D j gives S x (0) = X j =0 ( − j X m x ( m, D j )=1 λ ( m ) χ ( m ) ≪ A,K x (log x ) − K . (5.3)In particular, letting B = 11 A we have S h ( x ) = S x + h (0) − S x (0) ≪ A x (log x ) B ≪ A h (log x ) B , where h ≍ x (log x ) − B , and so J = 1 Y Z YY (cid:12)(cid:12)(cid:12) h S h ( x ) (cid:12)(cid:12)(cid:12) d x ≪ (log X ) − B + 1 Y Z YY (cid:12)(cid:12)(cid:12)(cid:12) h S h ( x ) − h S h ( x ) (cid:12)(cid:12)(cid:12)(cid:12) d x . (5.4)Then Lemma 4.6 (Parseval) with T = (log X ) B and h = Y /T = Y / (log X ) B gives J ≪ (log X ) − B + Z Y/h T | G (1 + it ) | d t + max T > Y/h Y /h T Z TT | G (1 + it ) | d t . (5.5) ow for the latter integral over [ T, T ], we apply the Lemma 4.1 (mean value) if T > X/ T ∈ [ Y /h , X/ J ≪ (cid:0) Y/h Y/Q + 1 (cid:1) (log X ) − B + max T > X/ Y /h T ( T /X + 1)+ max Y/h T X/ Y /h T (cid:0) Q TY + 1 (cid:1) (log X ) − B ≪ ( Q h + 1)(log X ) − B + Yh X ≪ W (log X ) − B = (log X ) A − B . (5.6)Here we used h > H/W , Q = H/W , (and Y ∈ [ X/W , X ]). Hence recalling B = 11 A gives Proposition 3.4 as claimed. (cid:3) Mean value of Dirichlet polynomials. In this subsection, we prove Proposition 5.1.Recall the definitions (2.4), (2.9),[ P , Q ] = [(log X ) A , (log X ) ψ ( X ) − A ] , [ P , Q ] = [exp (cid:0) (log X ) / δ/ (cid:1) , exp (cid:0) (log X ) − δ/ (cid:1) ] , S d ( X, A, δ ) = { m X/d : ∃ p , p | m with p j ∈ [ P j , Q j ] } . Let B = 11 A , and α = 1 / 5. Let V = P / = (log X ) B and define the prime polynomial Q v,j ( s ) := X P j p Q j e v/V p e ( v +1) /V λ ( p ) χ ( p ) p s . (5.7)Note Q v,j ( s ) = 0 only if v ∈ I j := { v ∈ Z : P j e v/V Q j } = [ ⌊ V log P j ⌋ , V log Q j ].We decompose [ T , T ] = T ∪ T as a disjoint union, where T = [0 , \ T and T = { t : | Q v, (1 + it ) | e − αv/V ∀ v ∈ I } . (5.8)For j = 1 , S ( j ) d the integers containing a prime factor in the interval [ P i , Q i ] with i = j and possibly, but not necessarily, with i = j . That is, S ( j ) d = { m X/d : ∃ p | m with p ∈ [ P i , Q i ] for i = j } . Also define the polynomial R v,j ( s ) = X Y e − v/V m Y e − v/V m ∈S ( j ) d λ ( m ) χ ( m ) m s · { p | m : P j p Q j } + 1 . (5.9)Now Lemma 4.7 (Ramar´e) applies with V = V, P = P j , Q = Q j and a m = λ χ S ( m ), c p = λ χ ( p ), b m = λ χ S j ( m ), giving Z T j | G (1 + it ) | d t ≪ V log Q j X v ∈I j Z T j | Q v,j (1 + it ) R v,j (1 + it ) | d t + 1 V + 1 P j + X Y n Yp ∤ n ∀ p ∈ [ P j ,Q j ] S d ( n ) n . e crucially note the latter sum vanishes since each n ∈ S d has a prime factor p ∈ [ P j , Q j ].Summing over j = 1 , 2, the second and third terms above contribute ≪ X j (cid:16) V + 1 P j (cid:17) ≪ V = (log X ) − B . Hence the desired integral is Z TT | G (1 + it ) | d t = Z T ∪T | G (1 + it ) | d t ≪ E + E + (log X ) − B , (5.10)where E j = V log Q j X v ∈I j Z T j | Q v,j (1 + it ) R v,j (1 + it ) | d t . (5.11)Hence it suffices to bound E , E ≪ ( Q T /Y + 1)(log X ) − B . Bound for E : By definition of t ∈ T , we have | Q v, (1 + it ) | e − αv/V for all v ∈ I , so E ≪ V log Q X v ∈I e − αv/V Z T | R v, (1 + it ) | d t ≪ V log Q X v ∈I e − αv/V (cid:16) TY /e v/V + 1 (cid:17) by Lemma 4.1 (mean value). Summing the resulting geometric series gives E ≪ V log Q P − α − e − α/V (cid:16) Q TY + 1 (cid:17) ≪ (log X ) − α (cid:16) Q TY + 1 (cid:17) ≪ (log X ) − B (cid:16) Q TY + 1 (cid:17) , (5.12)noting V / (1 − e − α/V ) = O (1) and 1 − B/ < − B . Bound for E : We choose the maximizing v ∈ I for E . Thus since |I | < V log Q , E = V log Q X v ∈I Z T | Q v, · R v, (1 + it ) | d t ≪ ( V log Q ) Z T | Q v, · R v, (1 + it ) | d t ( V log Q ) X n sup t n ∈ [ n,n +1] ∩ T | Q v, · R v, (1 + it n ) | V log Q ) X t ∈W | Q v, · R v, (1 + it ) | , for a well-spaced set W ⊂ T . For instance, one may take W as the even or odd integers in T (choose the parity that gives a larger contribution). We shall see W is easier to analyzethan T itself.Now is the critical step for the choice g = λχ and log P = (log X ) / δ : by Lemma 4.5(Vinogradov-Korobov), we have for all t ∈ [ T , T ] | Q v, (1 + it ) | ≪ δ,A log X T + (log X ) − B ≪ (log X ) − B , (5.13) or T = (log X ) B and B = 11 A . So by Lemma 4.3 (Hal´asz-Montgomery), we have E ≪ ( V log Q ) (log X ) − B X t ∈W | R v, (1 + it ) | ≪ ( V log Q ) (log X ) − B ( Y e − v/V + |W|√ T ) e v/V Y ≪ (log X ) − B (1 + |W| √ T Q Y ) , recalling log T ≍ log X , V = (log X ) B , and e v/V Q .Thus it suffices to bound |W| . We shall obtain E ≪ (log X ) − B (cid:16) TY (cid:17) ≪ A,δ (log X ) − B (5.14)provided we show |W| ≪ T / /Q . To prove this, by definition of T ⊃ W , we first partition W = S u ∈I W ( u ) where | Q u, (1 + it ) | > e − uα/V for all t ∈ W ( u ) . Hence for each u ∈ I , we may apply Lemma 4.4 to the prime polynomial Q u, with U = e uα/V and L = e u/V , so that |W| = X u ∈I |W ( u ) | ≪ X u ∈I U T log U +log log T log L ≪ |I | U T α + T log L ≪ T / / A + o (1) , (5.15)since |I | < V log Q ≪ T o (1) , U Q α ≪ T o (1) , log L > log P > A log log T , by recalling[ P , Q ] = [(log X ) A , (log X ) ψ ( X ) − A ] and T ∈ [ X / , X ].Hence A > |W| ≪ T / /Q , and completes the proof of Proposition 5.1.6. Average Chowla-type correlations In this section, we establish the results for higher correlations stated in the introduction.We first exhibit quantitative cancellation among a broad class of correlations with a ‘typ-ical’ factor S µ . We use a standard ‘van der Corput’ argument and then apply the keyFourier estimate. Lemma 6.1. Given any A > , δ > , let S = S ( X, A, δ ) as in (2.5) . Assume H < X satisfies log H/ log X → ∞ , and G : N → C satisfies P n X | G ( n ) | ≪ X (log X ) A/ . Then X h H (cid:12)(cid:12)(cid:12) X n X S µ ( n + h ) G ( n ) (cid:12)(cid:12)(cid:12) ≪ A,δ HX (log X ) A/ . Proof. Let g = S µ . By Cauchy-Schwarz it suffices to show HX (log X ) A/ ≫ X h H (cid:12)(cid:12)(cid:12)(cid:12) X n X g ( n + h ) G ( n ) (cid:12)(cid:12)(cid:12)(cid:12) = X n,n ′ X G ( n ) G ( n ′ ) X h H g ( n + h ) g ( n ′ + h ) . Using Cauchy-Schwarz again, the right hand side above is bounded by X n X | G ( n ) | · (cid:18) X n,n ′ (cid:12)(cid:12)(cid:12) X h H g ( n + h ) g ( n ′ + h ) (cid:12)(cid:12)(cid:12) (cid:19) . ecalling g is supported on [1 , X ]. By assumption P n X | G ( n ) | ≪ X (log X ) A/ , so itsuffices to prove H X (log X ) A/ ≫ X n,n ′ (cid:12)(cid:12)(cid:12) X h H g ( n + h ) g ( n ′ + h ) (cid:12)(cid:12)(cid:12) = X | h | Given any A > , δ > , let S = S ( X, A, δ ) as in (2.5) . Assume H < X satisfies log H/ log X → ∞ , and G : N → C satisfies P n X | G ( n ) | ≪ X (log X ) A/ . Then X h ,..,h m H (cid:12)(cid:12)(cid:12)(cid:12) X n X G ( n ) m Y j =1 µ ( n + h j ) (cid:12)(cid:12)(cid:12)(cid:12) (6.1) ≪ A,δ X h ,..,h m H X n X | G ( n ) | m Y j =1 S ( n + h j ) + mXH m (log X ) A/ . Proof. We observe from Lemma 6.1 that any correlation with a factor S µ exhibits strongcancellation. So we split up µ = S µ + S µ until each term has a factor S µ , except for oneterm with m factors of S µ . Thus the product in (6.1) becomes m Y j =1 µ ( n + h j ) = m Y j =1 S µ ( n + h j ) + m X i =1 S µ ( n + h i ) Y j
Proof of Theorem 1.3. Let G ( n ) = Q kj =1 Λ( n + a j ) for the tuple A = { a , .., a k } . Then P n X | G ( n ) | ≪ X (log X ) k , and by a standard sieve upper bound X n X G ( n ) m Y j =1 S ( n + h j ) ≪ m, A X (cid:16) Y p ∈ [ P ,Q ] + Y p ∈ [ P ,Q ] (cid:17)(cid:16) − p (cid:17) m ≪ m,δ, A Xψ δ ( X ) m , using Mertens’ product theorem. Hence Theorem 6.2 with A = 20( m + k ) gives X h ,..,h m H (cid:12)(cid:12)(cid:12)(cid:12) X n X k Y j =1 Λ( n + a j ) m Y j =1 µ ( n + h j ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ m,δ, A XH m ψ δ ( X ) m . (6.4) (cid:3) Proof of Theorem 1.6. Let G ( n ) = Q ji =1 d k i ( n + a i ) for the tuple A = { a , .., a j } and recall k = P ji =1 k i . Using work of Henriot [5, Theorem 3], we may obtain X h H X n X S ( n + h ) j Y i =1 d k i ( n + a i ) ≪ A HX (log X ) j +1 X n √ X S ( n ) n j Y i =1 X n √ X d k i ( n ) n . By the divisor bound P n √ X d k i ( n ) /n ≪ X (log X ) k i , and by Mertens’ product theorem X n √ X S ( n ) n ≪ log X (cid:16) Y p ∈ [ P ,Q ] + Y p ∈ [ P ,Q ] (cid:17)(cid:16) − p (cid:17) ≪ δ log Xψ δ ( X ) . Thus since P n X | G ( n ) | ≪ X (log X ) k , Theorem 6.2 with A = 20 k gives X h H (cid:12)(cid:12)(cid:12)(cid:12) X n X µ ( n + h ) j Y i =1 d k i ( n + a i ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ δ, A HXψ δ ( X ) (log X ) k − j . (cid:3) Almost all shifts. Corollary 1.5 follows from the following result by the triangle in-equality for g j = µ . Theorem 6.3. Suppose log H/ log X → ∞ as X → ∞ . Let g = µ and take any g j : N → C with | g j | for < j k . Then for any fixed shifts h , ..., h k H , K > we have X p X k Y j =1 g j ( p + h j ) = o ( π ( X )) , (6.5) for all except O K ( H (log X ) − K ) shifts h H .Proof. Given ε > h , .., h k H , we aim to show |E | ≪ ε H (log X ) − K for theexceptional set E = n h H : (cid:12)(cid:12)(cid:12) X p X µ ( p + h ) k Y j =2 g j ( p + h j ) (cid:12)(cid:12)(cid:12) > επ ( X ) o . (6.6) o this, by Markov’s inequality we have |E | ( επ ( X )) ≪ X h ∈E (cid:12)(cid:12)(cid:12)(cid:12) X p X µ ( p + h ) k Y j =2 g j ( p + h j ) (cid:12)(cid:12)(cid:12)(cid:12) X h ∈E (cid:12)(cid:12)(cid:12)(cid:12) X p X S ( p + h ) (cid:12)(cid:12)(cid:12)(cid:12) + X h H (cid:12)(cid:12)(cid:12)(cid:12) X p X S µ ( p + h ) k Y j =2 g j ( p + h j ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ A π ( X ) ψ ( X ) X h ∈E Y p | hp>P (1 + p ) + Hπ ( X )(log X ) A/ , using Proposition 6.1 when p + h ∈ S , and a standard sieve upper bound [2, Theorem 7.1]when p + h / ∈ S . Here S = S ( X, A, δ ) as in (2.5) with A = 80 K and δ = 1 / 10, say.Observe for any h H = (log X ) ψ ( X ) the above product is at most Q P P (1 + p ) = o (cid:0) |E | π ( X ) (cid:1) . Hence we conclude |E | ≪ ε H (log X ) − K as desired. (cid:3) Non-pretentious multiplicative functions In this section we prove Theorem 1.8, which we restate below. Theorem 1.8 Let H = X θ for θ ∈ (0 , , and take a multiplicative function f : N → C with | f | . There exists ρ ∈ (0 , ) such that, if M ( f ; X /H − ρ , Q ) → ∞ as X → ∞ for eachfixed Q , then X h H (cid:12)(cid:12)(cid:12) X p X f ( p + h ) (cid:12)(cid:12)(cid:12) = o θ,ρ (cid:0) Hπ ( X ) (cid:1) . Proof. Consider the exponential sum F x ( α ) = P x m x + H f ( m ) e ( mα ). The hypotheses ofour theorem are made in order to satisfy [12, Theorem 1.4], which in this case gives Z X sup α | F x ( α ) | d x = o θ,ρ ( HX ) . (7.1)We critically note the supremum is inside the integral.Now on to the proof, it suffices to show S f = o ( HX ) where S f := X h H (cid:12)(cid:12)(cid:12) X n X Λ( n ) f ( n + h ) (cid:12)(cid:12)(cid:12) ≪ X h H (2 H − h ) (cid:12)(cid:12)(cid:12) X n X Λ( n ) f ( n + h ) (cid:12)(cid:12)(cid:12) . e introduce coefficients c ( h ) to denote the phase of P n X Λ( n ) f ( n + h ), so that S f ≪ H X h H ( H − h ) c ( h ) X n X Λ( n ) f ( n + h )= 1 H X h H c ( h ) X n X Λ( n ) X m X + H f ( m ) m = n + h · Z X x n,m x + H d x = 1 H Z X Z X h H c ( h ) e ( hα ) X x n,m x + H Λ( n ) f ( m ) e (cid:0) ( n − m ) α (cid:1) d α d x , by orthogonality R e ( nα ) d α = n =0 . That is, we have the following triple convolution S f ≪ H Z X Z C ( α ) L x ( − α ) F x ( α ) d α d x , (7.2)denoting the sums C ( α ) = P h H c ( h ) e ( hα ) and L x ( α ) = P x n x + H Λ( n ) e ( nα ).We shall split the inner integral on α according to the size of L x . Specifically, for each x let T x = { α ∈ [0 , 1] : | L x ( α ) | > δH } . Then by Markov’s inequality, T x has measure Z T x d α δH ) Z T x | L x ( α ) | d α ≪ δ H , (7.3)since the Fourier identity implies Z | L x ( α ) | d α = X x n ,n ,n ,n x + H Λ( n )Λ( n )Λ( n )Λ( n ) n + n = n + n ≪ (log X ) X x p ,p ,p ,p x + Hp + p = p + p ≪ θ H , by a standard sieve upper bound [2, Theorem 7.1]. Thus plugging (7.3) into (7.2), we obtain S f ≪ H Z X Z [0 , \T x C ( α ) L x ( − α ) F x ( α ) d α d x + 1 δ H Z X sup α ∈T x | C ( α ) L x ( − α ) F x ( α ) | d x . Denote the two integrals above by I and I . Observe I ≪ θ δ − R X sup α | F x ( α ) | d x ,using | C ( α ) | H trivially and | L x ( α ) | ≪ θ H by the Brun–Titchmarsh theorem. Then bydefinition of T x , Cauchy-Schwarz implies I δ Z X Z [0 , \T x | C ( α ) F x ( α ) | d α d x δ Z X (cid:18) Z | C ( α ) | d α · Z | F x ( α ) | d α (cid:19) / d x δHX, by Parseval’s identity applied to C and F x . 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