Bombieri-Vinogradov for multiplicative functions, and beyond the x 1/2 -barrier
aa r X i v : . [ m a t h . N T ] A p r BOMBIERI-VINOGRADOV FOR MULTIPLICATIVEFUNCTIONS, AND BEYOND THE x / -BARRIER ANDREW GRANVILLE AND XUANCHENG SHAO
Abstract.
Part-and-parcel of the study of “multiplicative num-ber theory” is the study of the distribution of multiplicative func-tions in arithmetic progressions. Although appropriate analogiesto the Bombieri-Vingradov Theorem have been proved for partic-ular examples of multiplicative functions, there has not previouslybeen headway on a general theory; seemingly none of the differentproofs of the Bombieri-Vingradov Theorem for primes adapt wellto this situation. In this article we find out why such a result hasbeen so elusive, and discover what can be proved along these linesand develop some limitations. For a fixed residue class a we extendsuch averages out to moduli ≤ x − δ . Formulating the Bombieri-Vingradov Theorem formultiplicative functions
The Bombieri-Vingradov Theorem, a mainstay of analytic numbertheory, shows that the prime numbers up to x are “well-distributed”in all arithmetic progressions mod q , for almost all integers q ≤ x / − ε .To be more precise, for any given sequence f (1) , f (2) , . . . , we define∆( f, x ; q, a ) := X n ≤ xn ≡ a (mod q ) f ( n ) − ϕ ( q ) X n ≤ x ( n,q )=1 f ( n ) . The Bombieri-Vinogradov Theorem states that if f is the characteristicfunction for the primes, then for any given A > B = Date : April 22, 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Multiplicative functions, Bombieri-Vinogradov theorem,Siegel zeroes.A.G. has received funding from the European Research Council grant agreementn o B ( A ) > Q ≤ x / / (log x ) B then(1.1) X q ∼ Q max a : ( a,q )=1 | ∆( f, x ; q, a ) | ≪ x (log x ) A , where, here and henceforth, “ q ∼ Q ” denotes the set of integers q inthe range Q < q ≤ Q . The analogous result is known to hold when f = µ , the Mobius function, and when f is the characteristic functionfor the y -smooth numbers [16, 23], and the literature is swarming withmany other interesting examples besides. There are many proofs ofthe original Bombieri-Vinogradov Theorem: more modern proofs relyon bilinearity and Vaughan’s identity, for example see chapter 28 of[8], or the very elegant proof in Theorems 9.16, 9.17, and 9.18 of [18].The extraordinary generality of the latter proof leads one to guess thatsomething like the Bombieri-Vinogradov Theorem should be true formost sensible arithmetic functions f .In this paper we focus on proving a Bombieri-Vinogradov type The-orem for multiplicative functions f which take values within the unitcircle, something that has been proved for several interesting examples,and one might guess is true in some generality. However one needs tobe careful: if f ( n ) = ( n/ f iscertainly not well-distributed in arithmetic progressions mod 3, nor inarithmetic progressions mod q , whenever 3 divides q . More generallyif f = χ is a primitive character mod r , or even if f is “close” to χ ,then f is not well-distributed in arithmetic progressions mod q , when-ever r divides q . So this is a significant departure from the classicalBombieri-Vinogradov type Theorem, in that it seems likely that X q ∼ Q max a : ( a,q )=1 | ∆( f, x ; q, a ) | ≫ max χ (mod r ) χ primitive r> ϕ ( r ) | S f ( x, χ ) | , where S f ( x, χ ) := X n ≤ x f ( n ) χ ( n ) . Therefore if f strongly correlates with some χ of “small” conductor(that is, S f ( x, χ ) ≫ x/ (log x ) C for some non-principal character χ ofconductor ≪ (log x ) D , for for some fixed C, D >
0) then (1.1) cannothold for all A . So to prove something like (1.1) we need to assumethat no such character exists. The usual way to formulate this involvesequidistribution for moduli of small conductor (see e.g. (9.68) in [18]):For any given A > This is discussed in detail in section 3.4, and proved there up to a factor of log x . OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 3
The A -Siegel-Walfisz criterion : We say that f satisfies the A -Siegel-Walfisz criterion if X n ≤ xn ≡ a (mod q ) f ( n ) − ϕ ( q ) X n ≤ x ( n,q )=1 f ( n ) ≪ A x (log x ) A for all ( a, q ) = 1 and x ≥
2. We say that f satisfies the Siegel-Walfiszcriterion if this holds for any fixed A > F ( s ) = ∞ X n =1 f ( n ) n s and − F ′ ( s ) F ( s ) = ∞ X n =2 Λ f ( n ) n s , for Re( s ) >
1. Following [20], we restrict attention to the class C ofmultiplicative functions f for which | Λ f ( n ) | ≤ Λ( n ) for all n ≥ . This includes most multiplicative functions of interest, including all1-bounded completely multiplicative functions. Two key observationsare that if f ∈ C then each | f ( n ) | ≤
1, and if F ( s ) G ( s ) = 1 then g ∈ C . Corollary 1.1.
Let f be a multiplicative function with f ∈ C , andassume that f satisfies the -Siegel-Walfisz criterion. Fix δ, ε > . If Q ≤ x / − δ then X q ∼ Q max a : ( a,q )=1 | ∆( f, x ; q, a ) | ≪ x (log x ) − ε . This bound is considerably weaker than the hoped-for bound (1.1)in that we improve upon the “trivial bound”, ≪ x , by only a factor(log x ) − ε rather than by an arbitrary power of log x . However we willshow in Proposition 1.4 that Corollary 1.1 is, up to the ε -factor, bestpossible.1.1. Taking exceptional characters into account.
Even if the 1-Siegel-Walfisz criterion does not hold we can prove a version of theBombieri-Vinogradov Theorem which takes account of the primitivecharacters ψ for which S f ( x, ψ ) is large.For any character ψ (mod r ), define σ f ( x, ψ ) := sup x / Fix δ, ε > and let k be the largest integer ≤ /ε . Forany f ∈ C , and for any Q ≤ x / − δ , we have X q ∼ Q max a : ( a,q )=1 | ∆ k ( f, x ; q, a ) | ≪ x (log x ) − ε . Corollary 1.1 will follow from Theorem 1.2.Theorem 1.2 is close to “best possible” in that (as we show in sec-tion 8.4), for given integer k , there is an ε ′ ≍ √ k , for which there exists f ∈ C such that X q ∼ Q max a : ( a,q )=1 | ∆ k ( f, x ; q, a ) | ≫ x (log x ) − ε ′ . From this perspective, the usual formulation implicitly assumes that ψ = 1, whichis true whenever f ( n ) ≥ n . OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 5 In [20] it is proved that if log q ≤ (log x ) δ where δ = δ ( ε ) > 0, then(1.3) | ∆ k ( f, x ; q, a ) | ≪ ϕ ( q ) x (log x ) − ε whenever ( a, q ) = 1, where k is the largest integer ≤ /ε . That is, f is well-distributed in all arithmetic progressions mod q for all q ≤ Q provided x is very large compared to Q , and in Theorem 1.2 we haveshown good distribution in all arithmetic progressions mod q for almostall q ≤ Q provided x > Q δ , a much larger range for q but at the costof some possible exceptions. This is reminiscent of what we know aboutthe distribution of prime numbers in arithmetic progressions, thoughhere we have significantly weaker bounds on the error terms.1.2. Limitations on the possible upper bounds. One might guessthat if there are no characters of small conductor that strongly correlatewith f then perhaps one can significantly improve the upper boundsin Theorem 1.2 and Corollary 1.1 (if one assumes an A -Siegel-Walfiszcriterion for all A > f ( p ) for the large primes p up to x , do their utmostto block equi-distribution: Proposition 1.3. Let g be a multiplicative function with each | g ( n ) | ≤ , and suppose we are given Q ≤ x with Q, x/Q → ∞ as x → ∞ . Thereexists a subset P of the primes in the interval ( x/ , x ] , that containsalmost all of those primes, and a constant σ ∈ {− , } , such that if f ( n ) = g ( n ) for all n ≤ x other than n ∈ P , and f ( p ) = σ for all p ∈ P , then | ∆( f, x ; q, | ≫ π ( x ) /ϕ ( q ) for at least half of the moduli q ∼ Q . This implies that (1.4) X q ∼ Q | ∆( f, x ; q, | ≫ x log x . It is widely believed that for any fixed ε, A > π ( x ; q, a ) = π ( x ) ϕ ( q ) (cid:18) O (cid:18) x ) A (cid:19)(cid:19) whenever ( a, q ) = 1 and q ≤ x − ε .We now show that even assuming a strong Siegel-Walfisz criterion,we expect that one cannot significantly improve the upper bound inCorollary 1.1. The definition of ∆ k ( f, x ; q, a ) in [20] involves removing the k largest charactersums for characters mod q , whereas here we use the characters mod q induced fromthe largest character sums for characters mod r , for any r | q with r ≤ log x . This isa minor technical difference, which is sorted out in Corollary 3.2. Here “almost all of the primes” in ( x/ , x ] means (1 + o (1))( π ( x ) − π ( x/ A. GRANVILLE AND X. SHAO Proposition 1.4. Assume (1.5) . Let x / < Q < x/ . There existsa completely multiplicative function f , taking only values − and ,which satisfies the A -Siegel-Walfisz criterion for all A > , but forwhich (1.4) holds. We have exhibited two fundamental obstructions to a Bombieri-Vinogradov Theorem for multiplicative functions:(i) If f correlates closely with a character of small conductor; or(ii) If the values of f ( p ) with x/ < p ≤ x conspire against equi-distribution.Consequently, although we have been able to beat the “trivial bound”by a factor of (log x ) − ε by taking (i) into account in Theorem 1.2, (ii)ensures that we cannot do much better in general.These two obstructions have arisen before in the multiplicative func-tions literature, in Montgomery and Vaughan’s seminal work [27] onbounding exponential sums twisted by multiplicative coefficients. Forthis question, the contributions from obstruction (i) are identified pre-cisely in [6], but the sharpness of the bounds are inevitably restrictedby obstruction (ii). However, in Proposition 1 of [5], de la Bret`echeshowed that one can obtain much better bounds if one restricts at-tention to f that are supported only on smooth numbers, since thenobstruction (ii) is rendered irrelevant. We can do much the same here:1.3. Multiplicative functions supported on smooth numbers. The following key result generalizes (1.3) (which was proved in [20])to error terms in which one “saves” an arbitrary power of log x , formultiplicative functions supported on smooth numbers.Given a finite set of primitive characters, Ξ, let Ξ q be the set ofcharacters mod q that are induced by the characters in Ξ, and thendefine ∆ Ξ ( f, x ; q, a ) := X n ≤ xn ≡ a (mod q ) f ( n ) − ϕ ( q ) X χ ∈ Ξ q χ ( a ) S f ( x, χ ) . Proposition 1.5. Fix ε > and < γ ≤ − ε , let y = x γ , andsuppose that f ∈ C , and is only supported on y -smooth numbers. Fix B ≥ . There exists a set, Ξ , of primitive characters ψ (mod r ) witheach r ≤ R := x ε/ (3 log log x ) , containing ≪ (log x ) B +7+ o (1) elements, Smooth numbers are integers with no large prime factors. That is, we restrictattention to multiplicative function f ( . ) for which f ( p k ) = 0 for any prime p > y and integer k ≥ OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 7 such that if q ≤ R and ( a, q ) = 1 then | ∆ Ξ ( f, x ; q, a ) | ≪ ϕ ( q ) x (log x ) B . By (1.3) we know that if B < f ∈ C (not just those supported on y -smooth numbers) with the sizeof | Ξ | bounded only in terms of B .We state two Bombieri-Vinogradov type Theorems that follow fromthis. Theorem 1.6. Fix < ε ≤ , A, B with B > and A > B + 7 .Let y = x ε . Let f ∈ C be a multiplicative function which is onlysupported on y -smooth integers. There exists a set, Ξ , of primitivecharacters, containing ≪ (log x ) B +7+ o (1) elements, such that for any Q ≤ x / /y / (log x ) A , we have (1.6) X q ∼ Q max ( a,q )=1 | ∆ Ξ ( f, x ; q, a ) | ≪ B x (log x ) B . Corollary 1.7. Fix < ε ≤ . Let y = x ε . Let f ∈ C be a multiplica-tive function which is only supported on y -smooth integers. Assumethat the Siegel-Walfisz criterion holds for f . For any given B > there exists A such that for any Q ≤ x / / ( y / (log x ) A ) , we have X q ∼ Q max ( a,q )=1 | ∆( f, x ; q, a ) | ≪ B x (log x ) B . Note that these bounds are non-trivial since the number of x ε -smoothintegers up to x is ≫ x . Combining Corollary 1.7 with the machinerydeveloped in [29], one may prove for such multiplicative functions thattheir higher Gowers U k -norms are o (1) in progressions on average. Thisresult will be stated and discussed in Section 9.1.4. Breaking the x / -barrier. The main method used in our proofsis a modification of that developed by Green in [21]; see also [29] forusing a similar argument to deal with higher Gowers norms. Greenproved (a more general result which implies) that X q ∼ Qq prime | ∆( f, x ; q, | ≪ x log log x (log x ) , for any Q < x − ε , remarkably breaking the x / -barrier. In com-parison, previous results concerning breaking the x / -barrier in theoriginal Bombieri-Vinogradov theorem typically only work when q isrequired to be “smooth” [2, 15, 34], or else only beat the x / -barrier A. GRANVILLE AND X. SHAO by x o (1) [3, 4]. See also [9] and the references therein for results alongthe same line when f is the indicator function of smooth numbers.In Green’s result, the issue of correlations with non-trivial charac-ters of small conductor does not arise since no such character inducesa character modulo a large prime (and Green is only summing overprime moduli). Nonetheless obstruction (ii) still applies and so Propo-sition 1.3, as well as the construction in section 8.2, shows that Green’sresult is more-or-less best possible (up to the log log x factor). One canmodify Green’s proof to include composite moduli by taking accountof the characters ψ j , as we have done here. This leads to the followingextensions (for fixed a ) of Corollary 1.1 and Theorems 1.2, as well asTheorem 1.6 and Corollary 1.7. Theorem 1.8. Let f be a multiplicative function with f ∈ C . Fix δ, ε > and let k be the largest integer ≤ /ε . For any ≤ | a | ≪ Q ≤ x − δ , we have X q ∼ Q ( a,q )=1 | ∆ k ( f, x ; q, a ) | ≪ x (log x ) − ε . If f satisfies the -Siegel-Walfisz criterion then X q ∼ Q ( a,q )=1 | ∆( f, x ; q, a ) | ≪ x (log x ) − ε . Theorem 1.9. Fix δ, B > . Let y = x ε for some ε > sufficientlysmall in terms of δ . Let f ∈ C be a multiplicative function which isonly supported on y -smooth integers. Then there exists a set, Ξ , ofprimitive characters, containing ≪ (log x ) B +7+ o (1) elements, such thatfor any ≤ | a | ≪ Q ≤ x − δ , we have X q ∼ Q ( a,q )=1 | ∆ Ξ ( f, x ; q, a ) | ≪ x (log x ) B . If f satisfies the Siegel-Walfisz criterion then (1.7) X q ∼ Q ( a,q )=1 | ∆( f, x ; q, a ) | ≪ x (log x ) B . The proofs of these last two results, which break the x / -barrier, relyon a deep estimate of Bettin and Chandee [1] on bilinear Kloostermansums, which is an impressive development going beyond the famousestimates of Duke, Friedlander and Iwaniec [11]. It should be notedthat Fouvry and Tenenbaum (Th´eor`eme 2 in [17]) established (1.7) OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 9 unconditionally when f is the characteristic function of the y -smoothintegers for any Q ≤ x / − δ , and any y ≤ x ǫ .Our focus in this last part of the paper is to go beyond the x / -barrier by incorporating the necessary expedient of x ε -smooth functionsinto our arguments. Can one go much further beyond the x / -barrierusing current technology, especially if f is y -smooth for y a lot smallerthan x ε ? In a sequel to this paper, joint with Sary Drappeau, we willextend Theorem 1.9 to the range Q ≤ x / − ε and with a wide rangefor the smoothness parameter y , by incorporating somewhat differenttechniques into our arguments, and improving the x in the upper boundto Ψ( x, y ), the counting function for the y -smooths.Probably the most novel part of our work is to compare the meanvalue of f in the arithmetic progression a (mod q ) with the the meanvalue of f in the arithmetic progression a (mod q s ), where q s is thelargest w -smooth divisor of q . See, for example, Theorem 4.2.There is a series of seven papers by Elliott on multiplicative functionsin arithmetic progressions, some of which explore Bombieri-Vinogradovtype theorems (particularly [12, 13]). Several of the themes in thispaper have their origins in his seminal work. Notation. Let w = w ( x ) be a parameter, which will typically be afixed power of log x . For any positive integer q , we have a uniquedecomposition q = q s q r of q into a w -smooth part q s and a w -rough part q r , where q s = ( q, Q p ≤ w p ∞ ) is the largest w -smooth integer dividing q , and q r = q/q s has no prime factors ≤ w . Although the values of q r , q s depend on the parameter w , we will not explicitly indicate thisdependence as the choice of w should always be clear from the context.2. Smooth number estimates We call n a y - smooth integer if all of its prime factors are ≤ y . Welet P ( n ) denote the largest prime factor of n so that n is y -smooth ifand only if P ( n ) ≤ y .We need several well-known estimates involving the distribution ofsmooth numbers (unless otherwise referenced, see [19]). Let Ψ( x, y )be the number of y -smooth integers up to x . If y ≤ (log x ) o (1) thenΨ( x, y ) = x o (1) . Otherwise if x ≥ y ≥ (log x ) ε we write x = y u andthen Ψ( x, y ) = x/u u + o ( u ) . In particular if y = (log x ) A then Ψ( x, y ) = x − A + o (1) . Key conse-quences include if x ≥ y then X x 1, and thendetermine ρ ( u ) from the differential-delay equation ρ ′ ( u ) = − ρ ( u − /u for all u > 1. Then(2.2) Ψ( x, y ) = xρ ( u ) (cid:18) O (cid:18) log( u + 1)log y (cid:19)(cid:19) for x ≥ y ≥ exp((log log x ) ).Define α ( x, y ) to be the real number for which X p ≤ y log pp α − x ;one has y − α ≍ u log u when y ≫ log x . If x ≥ y ≥ (log x ) ε then α ≫ ε . We need the comparison bounds(2.3) Ψ( x/d, y ) = (cid:18) O (cid:18) u (cid:19)(cid:19) Ψ( x, y ) d α for d = y O (1) and, in general,(2.4) Ψ( x/d, y ) ≪ Ψ( x, y ) d · d − α which follow from Theorem 2.4 of [7]. This last bound implies that if y ≥ (log x ) ε then R x Ψ( t, y ) /t dt ≪ ε Ψ( x, y ). The bound in (2.3) isnot useful for us when u, d ≪ 1. In this range we have(2.5) Ψ( x/d, y ) = Ψ( x, y ) d α + O (1 / log x ) (cid:18) O (cid:18) log u log y (cid:19)(cid:19) by (2.22) and (2.23) of [7].Theorem 6 of [16] gives a version of the Bombieri-Vinogradov The-orem for y -smooth numbers (though see also [33], and see [23] for an OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 11 improvement on the range of y ): For any A > B = B ( A ) such that(2.6) X q ≤√ x/ (log x ) B max ( a,q )=1 (cid:12)(cid:12)(cid:12)(cid:12) Ψ( x, y ; q, a ) − Ψ q ( x, y ) ϕ ( q ) (cid:12)(cid:12)(cid:12)(cid:12) ≪ x (log x ) A where Ψ q ( x, y ) denotes the number of y -smooth integers up to x thatare coprime to q , and Ψ( x, y ; q, a ) denotes those that are ≡ a (mod q ).Then (4.11) of [7] gives the upper bound(2.7) Ψ q ( x, y ) ≪ ϕ ( q ) q Ψ( x, y )provided x ≥ y ≥ exp((log log x ) ) and q ≤ x .Corollary 2 of [24] implies a good upper bound from smooth numbersin short intervals: For any fixed κ > x + xT , y ) − Ψ( x, y ) ≪ κ Ψ( x, y ) T for 1 ≤ T ≤ min { y κ , x } . The contribution of characters Comparing large character sums for a primitive characterand the characters it induces. Recall the definition of σ f ( x, ψ ) fromthe introduction. We now define σ f ( x, z, ψ ) := sup x/z Suppose that f ∈ C . Let z ≥ exp((log log x ) ) and Q ≤ x .If χ (mod q ) is induced by ψ (mod r ) , where r ≤ q ≤ Q , then S f ( x, χ ) ≪ C xσ f ( x, z, ψ ) log log x + x (log x ) C , and S f ( x, ψ ) ≪ C xσ f ( x, z, χ ) log log x + x (log x ) C , for any given constant C > . Furthermore, for ψ (mod r ) with r ≤ Q we have X r | q ∼ Qχ mod q induced by ψ | S f ( x, χ ) | ϕ ( q ) ≪ C (cid:18) σ f ( x, z, ψ ) + 1(log x ) C (cid:19) xϕ ( r ) ; and if P ( r ) ≤ w then w X r | q ≤ Q, P ( q ) ≤ wχ mod q induced by ψ | S f ( x, χ ) | ϕ ( q ) ≪ C (cid:18) σ f ( x, z, ψ ) + 1(log x ) C (cid:19) xϕ ( r ) . Proof. Let h ( . ) be the multiplicative function which is supported onlyon the prime powers p k , for which p divides q but not r , with ( h ∗ f ψ )( p k ) = 0 for these p k . Thus h ∗ f ψ = f χ , and note that h ∈ C as f ∈ C , so that each | h ( m ) | ≤ 1. Now S f ( x, χ ) = X m ≤ x h ( m ) S f ( x/m, ψ )and therefore we obtain, as | S f ( x/m, ψ ) | ≤ σ f ( x, z, ψ ) x/m if m ≤ z , | S f ( x, χ ) | ≪ σ f ( x, z, ψ ) x X m ≤ zp | m = ⇒ p | q, p ∤ r m + x X z The first term in the upper bound is ≪ w X r | q ≤ QP ( q ) ≤ w ϕ ( q ) σ f ( x, z, ψ ) x X m ≤ zp | m = ⇒ p | q, p ∤ r m ≤ σ f ( x, z, ψ ) 1log w X r | q ≤ QP ( q ) ≤ w xϕ ( q ) X m ≤ zm | q, ( m,r )=1 µ ( m ) ϕ ( m ) ≤ σ f ( x, z, ψ ) x X m ≤ z ( m,r )=1 µ ( m ) ϕ ( m ) 1log w X mr | q ≤ QP ( q ) ≤ w ϕ ( q ) ≪ σ f ( x, z, ψ ) xϕ ( r ) X m ≤ z ( m,r )=1 µ ( m ) ϕ ( m ) w X n ≤ Q/mrP ( n ) ≤ w ϕ ( n ) ≪ σ f ( x, z, ψ ) xϕ ( r ) , writing q = mrn , and the claim follows. (cid:3) Focusing on large character sums. For fixed B > 0, let Ξ( B, Q )denote the set of primitive characters ψ (mod r ) with r ≤ Q for which σ f ( x, ψ ) ≥ x ) B . Corollary 3.2. Let f ∈ C and B > . (a) Suppose that Q ≤ x . If χ (mod q ) is a character with q ≤ Q and is not induced by any of thecharacters in Ξ( B, Q ) , then S f ( x, χ ) ≪ x log log x (log x ) B . (b) Now suppose that log Q = (log x ) o (1) and J ≥ is a given integerwith B < − / √ J . Then | Ξ( B, Q ) | < J and (3.1) | ∆ Ξ( B,Q ) ( f, x ; q, a ) | ≪ ϕ ( q ) x (log x ) B + o (1) for any q ≤ Q and ( a, q ) = 1 .Proof. (a) If χ is induced from ψ then σ f ( x, ψ ) ≤ / (log x ) B by thehypothesis, and the result then follows from the first part of Lemma3.1.(b) Suppose that there are at least J characters ψ j (mod r j ) inΞ( B, Q ). Let r = [ r , . . . , r J ] so that log r = (log x ) o (1) , and let χ j be the character mod r induced by ψ j , so that, for each j , there exists x / < X j ≤ x for which σ f ( X j , χ j ) ≫ / (log x ) B + o (1) by the second part of Lemma 3.1. However, by Theorem 6.1 of [20], one of these is ≪ / (log x ) − / √ J + o (1) , a contradiction.Now Theorem 6.1 of [20], applied to the set S of J − χ (mod q ) which give the J − | S f ( x, χ ) | , implies that | ∆ S ( f, x ; q, a ) | ≪ ϕ ( q ) x (log x ) − / √ J + o (1) . Write Ξ = Ξ( B, Q ). Now | S f ( x, χ ) | ≪ x/ (log x ) B + o (1) for every χ ∈ S \ Ξ q by (a), and also for every χ ∈ Ξ q \ S by the definition of S ,Theorem 6.1 of [20], and the hypothesis B < − / √ J . This impliesthat | ∆ Ξ ( f, x ; q, a ) − ∆ S ( f, x ; q, a ) | ≤ ϕ ( q ) X χ ∈ D | S f ( x, χ ) | ≪ ϕ ( q ) x (log x ) B + o (1) , where D is the symmetric difference of the sets S and Ξ q , and the resultfollows from adding the last two displayed equations. (cid:3) Corollary 3.3. Fix an integer J ≥ , then < B < − / √ J and let w = (log x ) B . For any f ∈ C we have w X q ≤ xP ( q ) ≤ w max ( a,q )=1 | ∆ Ξ( B, log x ) ( f, x ; q, a ) | ≪ x (log x ) B + o (1) . Moreover w X q ≤ xP ( q ) ≤ w max ( a,q )=1 | ∆ J − ( f, x ; q, a ) | ≪ x (log x ) B + o (1) . Here P ( q ) denotes the largest prime factor of q . Proof. Let Ξ = Ξ( B, log x ), which has no more than J elements byCorollary 3.2(b). We begin by bounding the contributions of the valuesof q > R := exp((log log x ) ): X R Let f ∈ C and Q ≤ x . For each q ∼ Q let a q (mod q ) be a residue class with ( a q , q ) = 1 . Suppose that Ξ is a set ofprimitive characters, containing ≪ (log x ) C elements, such that X q ∼ Q | ∆ Ξ ( f, x ; q, a q ) | ≪ x (log x ) B . If the D -Siegel-Walfisz criterion holds for f , where D ≥ B + C , then X q ∼ Q | ∆( f, x ; q, a q ) | ≪ x (log x ) B . Proof. By definition we have | ∆( f, x ; q, a q ) | ≤ | ∆ Ξ ( f, x ; q, a q ) | + 1 ϕ ( q ) X χ (mod q ) χ ∈ Ξ q , χ = χ | S f ( x, χ ) | . Summing this up over q ∼ Q , and using the hypothesis, we deduce that X q ∼ Q | ∆( f, x ; q, a q ) | ≤ X ψ ∈ Ξ ψ =1 X r ψ | q ∼ Qχ (mod q ) induced by ψ | S f ( x, χ ) | ϕ ( q ) + O (cid:18) x (log x ) B (cid:19) . The third part of Lemma 3.1 then implies that this is ≤ x X ψ ∈ Ξ ψ =1 ϕ ( r ) (cid:0) σ f ( x, ψ ) + (log x ) − D (cid:1) + O (cid:18) x (log x ) B (cid:19) . The D -Siegel-Walfisz criterion implies that for any non-principal ψ (mod r ), we have1 ϕ ( r ) S f ( X, ψ ) = 1 ϕ ( r ) X a (mod r ) ψ ( a )∆( f, X ; r, a ) ≪ X (log x ) D , for x / < X ≤ x , and so σ f ( x, ψ ) /ϕ ( r ) ≪ / (log x ) D . Therefore theabove is ≪ · x (log x ) D + x (log x ) B ≪ x (log x ) B provided D ≥ B + C . (cid:3) Lower bounds. If χ Ξ q then S f ( x, χ ) = X a (mod q ) χ ( a )∆ Ξ ( f, x ; q, a ) , and so | S f ( x, χ ) | ϕ ( q ) ≤ max ( a,q )=1 | ∆ Ξ ( f, x ; q, a ) | . Therefore max ( a,q )=1 | ∆ Ξ ( f, x ; q, a ) | ≥ max χ Ξ q | S f ( x, χ ) | ϕ ( q ) ;in particular we deduce that for any primitive ψ Ξ we have X q ∼ Q max ( a,q )=1 | ∆ Ξ ( f, x ; q, a ) | ≥ X q ∼ Qχ induced by ψ | S f ( x, χ ) | ϕ ( q ) . From the identity S f ( x, χ ) = X m ≤ x h ( m ) S f ( x/m, ψ )(see the proof of Lemma 3.1) we might expect that if | S f ( x, ψ ) | is largethen each | S f ( x, χ ) | should be too, though this is difficult to prove forevery induced χ . However we can do so when the smallest prime factorof q that does not divide r is > L log x , where L := x/ | S f ( x, ψ ) | . Taking OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 17 absolute values for such χ , and remembering the support of h ( . ), wehave | S f ( x, χ ) | ≥ | S f ( x, ψ ) | − X m> p | m = ⇒ p | q ( m,r )=1 | S f ( x/m, ψ ) | ≥ xL − X m> p | m = ⇒ p | q ( m,r )=1 xm ≥ xL − x Y p | q, p ∤ r (cid:18) − p (cid:19) − − ∼ xL = | S f ( x, ψ ) | , since q has o (log x ) prime factors, For such q we also have ϕ ( q ) ∼ ϕ ( r ) q/r . Therefore, if | S f ( x, χ ) | is significantly larger than ( x log x ) /Q then X q ∼ Qχ induced by ψ | S f ( x, χ ) | ϕ ( q ) ≫ | S f ( x, ψ ) | ϕ ( r ) X q ∼ Qq = rnp | n = ⇒ p>L log x n ≫ | S f ( x, ψ ) | ϕ ( r ) · L log x ) . Therefore if | S f ( x, ψ ) | ≫ x/ (log x ) A for some primitive ψ then(3.2) X q ∼ Q max ( a,q )=1 | ∆ Ξ ( f, x ; q, a ) | ≫ | S f ( x, ψ ) | ϕ ( r ) · x . At worst, when all of the | S f ( x, ψ ) | with ψ Ξ are small, we mightexpect (by orthogonality) that one has | S f ( x, ψ ) | ≫ S | f | ( x, ≫ √ x for some primitive character ψ (mod r ), from which one deduces that X q ∼ Q max ( a,q )=1 | ∆ Ξ ( f, x ; q, a ) | ≫ x max ψ (mod r ) ψ primitive ψ Ξ | S f ( x, ψ ) | ϕ ( r ) . Formulating the key technical result If χ (mod r ) is in Ξ, we write r = r χ , and note that it inducesa character mod q if and only if r divides q , and then the inducedcharacter is χξ q where ξ q is the principal character mod q . We havethe upper bound | S f ( x, χ ) | ≤ P n ≤ x, ( n,q )=1 ≪ ( ϕ ( q ) /q ) x for x ≥ q .Therefore | ∆ Ξ ( f, x ; q, a ) | ≪ | Ξ q | q x for x ≥ q . Moreover∆ Ξ ( f, x ; q, a ) = 1 ϕ ( q ) X χ (mod q ) χ Ξ q χ ( a ) S f ( x, χ ) . Corollary 4.1. Fix ε > , let Q ≤ x / − ε , and let w ≥ . Let Ξ be a set of primitive characters, each with w -smooth conductors ≤ Q/ exp((log w ) ) , such that (4.1) X χ ∈ Ξ r χ ≪ w / . For any -bounded multiplicative function f , we have X q ∼ Q max ( a,q )=1 | ∆ Ξ ( f, x ; q, a ) |≤ w X q s ≤ QP ( q s ) ≤ w max ( a,q s )=1 | ∆ Ξ ( f, x ; q s , a ) | + O (cid:18) xw / + x log log x log x (cid:19) . The proof can be modified to allow any Q ≤ x / / 2, though wewould need to replace the (log log x ) / log x by (log log x ) / log( x/Q ) onthe right-hand side.As we will justify below, Corollary 4.1 is a consequence of: Theorem 4.2. Fix ε > , let Q ≤ x / − ε , and let w ≥ . For any -bounded multiplicative function f , we have X q ∼ Q max ( a,q )=1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ xn ≡ a (mod q ) f ( n ) − q r X n ≤ xn ≡ a (mod q s ) f ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ xw / + x log log x log x , where q s , q r are the w -smooth and the w -rough parts of q , respectively. This will be proved in Section 4.2. Deduction of Corollary 4.1 from Theorem 4.2. Note that if ψ (mod r ) ∈ Ξ induces a character in Ξ q , then it also induces a character in Ξ q s , since r is w -smooth. It follows that∆ Ξ ( f, x ; q, a ) = X n ≤ xn ≡ a (mod q ) f ( n ) − ϕ ( q ) X ψ ∈ Ξ qs ψ ( a ) S f ( x, ψξ q ) , where ξ q is the principal character mod q . Combining this with thedefinition of ∆ Ξ ( f, x ; q s , a ) we arrive at the identity∆ Ξ ( f, x ; q, a ) = 1 q r ∆ Ξ ( f, x ; q s , a ) + X n ≤ xn ≡ a (mod q ) f ( n ) − q r X n ≤ xn ≡ a (mod q s ) f ( n )+ X ψ ∈ Ξ qs ψ ( a ) (cid:18) S f ( x, ψ ) − S f ( x, ψξ q ) ϕ ( q ) − (cid:18) ϕ ( q ) − q r ϕ ( q s ) (cid:19) S f ( x, ψ ) (cid:19) . OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 19 We sum the absolute value of this up over each q ∼ Q with a = a q which maximizes | ∆ Ξ ( f, x ; q, a ) | . The first term on the right-hand sideis then, summing up over q = q r q s ∼ Q , ≤ X q s ≤ QP ( q s ) ≤ w max ( a,q s )=1 | ∆ Ξ ( f, x ; q s , a ) | X q r ∼ Q/q s p | q r = ⇒ p>w q r ≪ w X q s ≤ QP ( q s ) ≤ w max ( a,q s )=1 | ∆ Ξ ( f, x ; q s , a ) | + X q s ∼ QP ( q s ) ≤ w max ( a,q s )=1 | ∆ Ξ ( f, x ; q s , a ) | . The last term comes from those q with q r = 1, in which case each | ∆ Ξ ( f, x ; q s , a ) | ≪ (1 + | Ξ q | ) · x/q s , and so X q s ∼ QP ( q s ) ≤ w max ( a,q s )=1 | ∆ Ξ ( f, x ; q s , a ) | ≪ X q s ∼ QP ( q s ) ≤ w xq s + X χ (mod r ) ∈ Ξ X q s ∼ QP ( q s ) ≤ wr | q s xq s ≪ xv − v + o ( v ) + X χ (mod r ) ∈ Ξ P ( r ) ≤ w xr X n ∼ Q/rP ( n ) ≤ w n ≪ X χ ∈ Ξ r χ ! xv − v + o ( v ) writing q s = rn in the second sum, as Q/r ≥ exp((log w ) ) = w v , say,and this term is ≪ x/w / by (4.1).By Theorem 4.2 we have X q ∼ Q max ( a,q )=1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ xn ≡ a (mod q ) f ( n ) − q r X n ≤ xn ≡ a (mod q s ) f ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ xw / + x log log x log x . Now since | S f ( x, ψ ) | ≤ P n ≤ x, ( n,q s )=1 ≪ ( ϕ ( q s ) /q s ) x , we have (cid:18) ϕ ( q ) − q r ϕ ( q s ) (cid:19) S f ( x, ψ ) ≪ (cid:18) q r ϕ ( q r ) − (cid:19) q r ϕ ( q s ) · ϕ ( q s ) q s x = (cid:18) q r ϕ ( q r ) − (cid:19) xq . Moreover, writing n = ab where p | a = ⇒ p | q and ( b, q ) = 1, we have (cid:12)(cid:12)(cid:12)(cid:12) S f ( x, ψ ) − S f ( x, ψξ q ) ϕ ( q ) (cid:12)(cid:12)(cid:12)(cid:12) = 1 ϕ ( q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ab ≤ xa> ( f ψ )( a )( f ψ )( b ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ϕ ( q ) X d | q r µ ( d ) ϕ ( d ) = X OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 21 Using Ramar´e’s weights. Let a q (mod q ) be the arithmeticprogression with ( a, q ) = 1 for which (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ xn ≡ a (mod q ) f ( n ) − q r X n ≤ xn ≡ a (mod q s ) f ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) is maximized, and then select ξ q with | ξ q | = 1 so that this equals ξ q X n ≤ xn ≡ a (mod q ) f ( n ) − q r X n ≤ xn ≡ a (mod q s ) f ( n ) . Therefore the left-hand side of the equation in Theorem 4.2 can bere-written as X n ≤ x f ( n ) F ( n )where, here and throughout, the function F is defined by(4.2) F ( n ) := X q ∼ Q ξ q (cid:18) n ≡ a q (mod q ) − q r n ≡ a q (mod q s ) (cid:19) . Sums like X n ≤ x f ( n ) F ( n )have been tackled in the literature using the Cauchy-Schwarz inequal-ity and then studying bilinear sums, such as what happens after (17) in[27]. Here we develop a formal inequality, which is a variant of Propo-sition 2.2 of [21]. The use of Ramar´e’s identity in this context waspioneered by Matom¨aki and Radziw l l [26]. Proposition 4.3. Let F : Z → C be an arbitrary function. Let ≤ Y < Z < x / be parameters, and write u = (log Z ) / log Y . Then forany -bounded multiplicative function f , we have X Z ≤ n ≤ x f ( n ) F ( n ) ≪ TY log Y + E sieve + E bilinear , where T = max d ≤ Z d X n ≤ xd | n | F ( n ) | ,E sieve = X n ≤ x | f ( n ) F ( n ) | ( n, Q Y ≤ p We use Ramar´e’s weight function: w ( n ) = 1 { Y ≤ p < Z : p | n } + 1 , to obtain the identity X Y ≤ p Therefore,(4.3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ≤ x f ( n ) F ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | Σ ′ | + E sieve + O (cid:18) TY log Y (cid:19) where Σ ′ := X m ≤ x/Y w ( m ) f ( m ) X Y ≤ p Y, Z ) for p , dyadically, intoΣ ′ ( P ) := X m ≤ x/P w ( m ) f ( m ) X P ≤ p< Pp ≤ x/m f ( p ) F ( pm ) OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 23 for P ∈ [ Y, Z ). Since F ( pm ) is supported on pm ≥ Z , we may addthe restriction m ≥ Z to the sum. By Lemma 2.1 of [21] we have X Z We nowapply Proposition 4.3 to the function F in (4.2) to prove Theorem 4.2.Let 2 ≤ Y < Z < x / be two parameters. Recall the notation that u = (log Z ) / log Y . Lemma 4.4 (Sieve term) . We have X n ≤ x | F ( n ) | ( n, Q Y ≤ p Using the trivial bound | F ( n ) | ≤ X q ∼ Q (cid:18) n ≡ a q (mod q ) + 1 q r n ≡ a q (mod q s ) (cid:19) , we may bound the desired expression by X q ∼ Q X n ≤ xn ≡ a q (mod q ) ( n, Q Y ≤ p Lemma 4.5 (Bilinear term) . For any P, Q ≥ we have E p,p ′ ∼ P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m ≤ min( x/p,x/p ′ ) F ( pm ) F ( p ′ m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ xP (cid:18) w log w + P . + log xπ ( P ) (cid:19) + Q , where p, p ′ denote primes, where, as usual, w ≤ x .Proof. By the definition of F , we change the order of summation towrite the inner sum over m asΣ( p, p ′ ) := X q,q ′ ∼ Q ξ q ξ q ′ X m ≤ min( x/p,x/p ′ ) K ( p, p ′ , q, q ′ ; m ) , where K ( p, p ′ , q, q ′ ; m ) is the expression (cid:18) pm ≡ a q (mod q ) − q r pm ≡ a q (mod q s ) (cid:19) (cid:18) p ′ m ≡ a q ′ (mod q ′ ) − q ′ r p ′ m ≡ a q ′ (mod q ′ s ) (cid:19) . The inner sum P m K ( p, p ′ , q, q ′ ; m ) can be written as a sum of foursums, the first of which is X m ≤ min( x/p,x/p ′ ) pm ≡ a q (mod q ) · p ′ m ≡ a ′ q (mod q ′ ) . This sum should have a main term of S ( p, p ′ , q, q ′ ) := ( min( x/p,x/p ′ )[ q,q ′ ] if ( p, q ) = ( p ′ , q ′ ) = 1 and p ′ a q ≡ pa ′ q (mod ( q, q ′ )) , OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 25 with an error of O (1). Similarly for the other three sums. It followsthat the sum X m ≤ min( x/p,x/p ′ ) K ( p, p ′ , q, q ′ ; m ) = g ( p, p ′ , q, q ′ ) + O (1) , where the main term g ( p, p ′ , q, q ′ ) is defined by g ( p, p ′ , q, q ′ ) = S ( p, p ′ , q, q ′ ) − q r S ( p, p ′ , q s , q ′ ) − q ′ r S ( p, p ′ , q, q ′ s )+ 1 q r q ′ r S ( p, p ′ , q s , q ′ s ) . The total contribution from the O (1) error is O ( Q ), which is accept-able. Thus it suffices to show that E p,p ′ ∼ P X q,q ′ ∼ Q | g ( p, p ′ , q, q ′ ) | ≪ xP (cid:18) w log w + P . + log xπ ( P ) (cid:19) . Note that we have the upper bound(4.4) | g ( p, p ′ , q, q ′ ) | ≪ xP · ( q, q ′ ) qq ′ . Moreover, when S ( p, p ′ , q, q ′ ) = 0 we have the (possibly) improved up-per bound(4.5) | g ( p, p ′ , q, q ′ ) | ≪ xP · ( q s , q ′ s ) qq ′ . Case 0. First consider the case when p | q or p ′ | q ′ . Then S ( p, p ′ , q, q ′ ) =0 and, so by (4.5), E p,p ′ ∼ P X q,q ′ ∼ Qp | q or p ′ | q | g ( p, p ′ , q, q ′ ) | ≪ xP π ( P ) X p,p ′ ∼ P X q,q ′ ∼ Qp | q or p ′ | q ( q s , q ′ s ) qq ′ , which by symmetry is ≪ xP π ( P ) X p ∼ P X q,q ′ ∼ Qp | q X d ≥ P ( d ) ≤ wd | q,d | q ′ dqq ′ = xP π ( P ) X p ∼ P p X d ≥ P ( d ) ≤ w d X r ∼ Q/pdr ′ ∼ Q/d rr ′ writing q = pdr, q ′ = dr ′ , which is ≪ xP π ( P ) · P · log w · ≪ x log wP , which is easily acceptable. Case 1. Now consider the case when ( p, q ) = ( p ′ , q ′ ) = 1 and p ′ a q ≡ pa ′ q (mod ( q, q ′ )). Since ( q s , q ′ ) = ( q, q ′ s ) = ( q s , q ′ s ), we have g ( p, p ′ , q, q ′ ) = min( x/p, x/p ′ ) (cid:18) ( q, q ′ ) qq ′ − ( q s , q ′ s ) qq ′ (cid:19) . It vanishes unless ( q r , q ′ r ) > 1, and thus by (4.4) it suffices to show thatΣ := 1 π ( P ) X q,q ′ ∼ Q ( q r ,q ′ r ) > X p,p ′ ∼ P ( p,q )=( p ′ ,q ′ )=1 p ′ a q ≡ pa ′ q (mod ( q,q ′ )) ( q, q ′ ) qq ′ ≪ w log w + P . + log xπ ( P ) . For fixed q, q ′ , p , the constraint p ′ a q ≡ pa q ′ (mod ( q, q ′ )) imposes a con-gruence condition on p ′ (mod ( q, q ′ )), and the number of p ′ satisfyingit is ≪ ( π ( P ) /ϕ (( q, q ′ )) if ( q, q ′ ) ≤ P . ,P/ ( q, q ′ ) + 1 if ( q, q ′ ) > P . . Here the bound in the first case follows from Brun-Titchmarsh, and inthe second case by dropping the primality condition on p ′ . Therefore(4.6) Σ ≪ Q X q,q ′ ∼ Q ( q r ,q ′ r ) > ( q, q ′ ) ϕ (( q, q ′ )) + 1 π ( P ) Q X q,q ′ ∼ Q ( q,q ′ ) >P . ( P + ( q, q ′ )) . In the second term let d = ( q, q ′ ) > P . and then the sum is ≤ X P . w p ≪ Q w log w . This completes the task of bounding Σ . Case 2. Finally consider the case when ( p, q ) = ( p ′ , q ′ ) = 1 and p ′ a q pa ′ q (mod ( q, q ′ )). If we further have p ′ a q pa ′ q (mod ( q s , q ′ s )), then S ( p, p ′ , q, q ′ ) = S ( p, p ′ , q s , q ′ ) = S ( p, p ′ , q, q ′ s ) = S ( p, p ′ , q s , q ′ s ) = 0 , and thus g ( p, p ′ , q, q ′ ) = 0. Hence we may impose the condition p ′ a q ≡ pa ′ q (mod ( q s , q ′ s )). By (4.5) it suffices to show thatΣ := 1 π ( P ) X q,q ′ ∼ Q X p,p ′ ∼ P ( p,q )=( p ′ ,q ′ )=1 p ′ a q pa ′ q (mod ( q,q ′ )) p ′ a q ≡ pa ′ q (mod ( q s ,q ′ s )) ( q s , q ′ s ) qq ′ ≪ w log w + P . + log xπ ( P ) . Note that the sum is nonempty only if ( q r , q ′ r ) > 1. Arguing as inCase 1, the number of p ′ satisfying the congruence condition on p ′ (mod ( q s , q ′ s )) is ≪ ( π ( P ) /ϕ (( q s , q ′ s )) if ( q s , q ′ s ) ≤ P . P/ ( q s , q ′ s ) + 1 if ( q s , q ′ s ) > P . . This leads to the upper boundΣ ≪ Q X q,q ′ ∼ Q ( q r ,q ′ r ) > ( q s , q ′ s ) ϕ (( q s , q ′ s )) + 1 π ( P ) Q X q,q ′ ∼ Q ( q s ,q ′ s ) >P . ( P + ( q s , q ′ s )) . Now as ( q s , q ′ s ) ≤ ( q, q ′ ) and ( q s ,q ′ s ) ϕ (( q s ,q ′ s )) ≤ ( q,q ′ ) ϕ (( q,q ′ )) , we bound Σ by thesame quantity with which we bounded Σ in (4.6), and the result fol-lows. (cid:3) Putting the pieces together. We now have the ingredients todeduce Theorem 4.2 from Proposition 4.3. Proof of Theorem 4.2. Recall that the left-hand side of the equation inTheorem 4.2 can be re-written as X n ≤ x f ( n ) F ( n )where the function F is defined as in (4.2). We bound this by ap-plying Proposition 4.3. Set Y = (log x ) and Z = x ε/ , so that u = log Z/ (log Y ) ≍ log x/ (log log x ). By Lemma 4.4 we have E sieve ≪ xu ≪ x log log x log x . By Lemma 4.5, the assumption Q ≤ x / − ε , and our choice of Y and Z , we have E bilinear ≪ x log x + x ( w log w ) / . To bound T , provided d ≤ Z ≤ x/Q we have X n ≤ xd | n | F ( n ) | ≤ X q ∼ Q X n ≤ xn ≡ a q (mod q ) d | n q r X q ∼ Q X n ≤ xn ≡ a q (mod q s ) d | n ≪ X q ∼ Q xqd ≪ xd , so that T ≪ x . The proof is completed by combining all these estimatestogether. (cid:3) Good error terms for smooth-supported f inarithmetic progressions In the following result we will prove a good estimate for all f sup-ported on y -smooth integers. In this article we will only use this with y a fixed power of x , but the full range will be useful in the sequel [10]. Proposition 5.1. Fix B ≥ and < η < . Given y = x /u in therange x / − η ≥ y ≥ exp(3( A + 1 + ǫ )(log log x ) / (log log log x )) , let R = R ( x, y ) := min { y log log log x u , x η x } ( ≤ y / o (1) ) . Suppose that f ∈ C , and is only supported on y -smooth numbers. Thereexists a set, Ξ , of primitive characters ψ (mod r ) with r ≤ R , con-taining ≪ u u + o ( u ) (log x ) B +7+ o (1) elements, such that if q ≤ R and ( a, q ) = 1 then | ∆ Ξ ( f, x ; q, a ) | ≤ ϕ ( q ) X χ (mod q ) χ Ξ q | S f ( x, χ ) | ≪ ϕ ( q ) Ψ( x, y )(log x ) B . OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 29 This immediately implies Proposition 1.5.Fix ε > 0. In Proposition 5.1 we let Ξ = Ξ(2 B + 2 + ε ), where wedefine Ξ( C ) to be the set of primitive characters ψ (mod r ) with r ≤ R such that there exists x η < X ≤ x for which(5.1) | S f ( X, ψ ) | ≥ Ψ( X, y )( u log u ) (log x ) C . We prove that Ξ( C ) has ≪ u u + o ( u ) (log x ) C +1 elements in section 5.11.5.1. A further support restriction for f . We will deduce Proposi-tion 5.1 from a similar (but stronger) result restricted to multiplicativefunctions f , which are only supported on those prime powers p k forwhich p ∈ ( qL, y ] where L ≥ q and L ≥ (log x ) C for an appropriatevalue of C .Fix a real number A . Let T ∗ q ( A ) = T q,f ( A, y ( q (log x ) A ) /α ) where T q,f ( A, z ) is the set of characters χ (mod q ) for which there exists x/z Fix A ≥ and < η < , and let x / − η ≥ y ≥ exp(3( A + 1)(log log x ) / (log log log x )) . Given q ≤ min { y / , x η/ } / (log x ) A +2 with ( a, q ) = 1 , let L = L q := q + (log x ) A +1 . If f ∈ C is only supported on prime powers p k with qL < p ≤ y , then (5.3) | ∆ T ∗ q ( A ) ( f, x ; q, a ) | ≤ ϕ ( q ) X χ (mod q ) χ ∗ q ( A ) | S f ( x, χ ) | ≪ ϕ ( q ) Ψ( x, y )(log x ) A . By minor modifications of the proof of Proposition 5.2 we will deducethe following: Corollary 5.3. Fix < η < and ε > , and let x / − η ≥ y ≥ exp(6(log log x ) / (log log log x )) . Given q ≤ min { y / , x η/ } with ( a, q ) = 1 , let L = L q := q + (log x ) .If f ∈ C is only supported on prime powers p k with qL < p ≤ y , then X χ (mod q ) | S f ( x, χ ) | ≪ Ψ( x, y )(log x ) ε . By Cauchying, the upper bound is ≪ q / Ψ( x, y ), so Corollary 5.3 isbetter in a wide range.In the next seven subsections we will prove Proposition 5.2. Through-out this section we will write ( u log u ) =: V for convenience, so that if1 ≤ d ≤ y then, as a consequence of (2.4),(5.4) Ψ( x/d, y ) ≪ V Ψ( x, y ) /d. Harper’s identity and Perron’s formula. Proof of Proposition 5.2. We use Harper’s identity: f ( n ) log n = Λ f ( n ) + Z ∞ β =0 n − β X abm = n Λ f ( a ) a β Λ f ( b ) f ( m ) dβ. Multiplying through by χ ( n ) and summing over n we obtain X x OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 31 We use Perron’s formula to try to work with the main term, applyingit at x and 2 x . Wlog we assume that m is only supported on ( x/y , x ].Therefore our integrand equals, for c = 1 + 1 / log x and x / ∈ Z ,12 iπ Z Re ( s )= c X x/y 2, which contribute ≪ xk log x X a,b ≤ y Λ( a ) a Λ( b ) b X xab (1 − k +1 T ) Let T q = T q ( A, x ) be the set of char-acters χ (mod q ) for which there exists x/y < X ≤ x , for which(5.5) | S f ( X, χ ) | > Ψ( X, y ) /V (log x ) A +2+ ε . Note that this is a weaker hypothesis than (5.2) in that the range for X is slightly shortened. Summing what remains of the main term overall χ T q , and dividing through by ϕ ( q ) we obtain the bound ≪ x Z s = c + it | t |≤ T Z ∞ β =0 ϕ ( q ) X χ T q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X a ≤ y Λ f ( a ) χ ( a ) a s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X b ≤ y Λ f ( b ) χ ( b ) b β + s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x/y Therefore our integral is V Ψ( x, y ) log x times ≪ Z s = c + itT ≥| t | >U Z ∞ β =0 ϕ ( q ) X χ (mod q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X a ≤ y Λ f ( a ) χ ( a ) a s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X b ≤ y Λ f ( b ) χ ( b ) b β + s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dβ ( x/y ) β dt | t | . We now use the Cauchy-Schwarz inequality, so that the square of thisintegral is ≤ Z s = c + itT ≥| t | >U Z ∞ β =0 ϕ ( q ) X χ (mod q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X a ≤ y Λ f ( a ) χ ( a ) a s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dβ ( x/y ) β dt | t | times Z s = c + itT ≥| t | >U Z ∞ β =0 ϕ ( q ) X χ (mod q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X b ≤ y Λ f ( b ) χ ( b ) b β + s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dβ ( x/y ) β dt | t | . We begin by noting that for s = c + it ,1 ϕ ( q ) X χ (mod q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X a ≤ y Λ f ( a ) χ ( a ) a s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X a,b ≤ y Λ f ( a )Λ f ( b )( ab ) c ( a/b ) it ϕ ( q ) X χ (mod q ) χ ( a ) χ ( b )= X a,b ≤ ya ≡ b (mod q ) Λ f ( a )Λ f ( b )( ab ) c ( a/b ) − it . Therefore, as | Λ f ( a )Λ f ( b ) / ( ab ) c | ≤ Λ( a )Λ( b ) /ab , and as f is only sup-ported on primes > qL , Z T | t | = U ϕ ( q ) X χ (mod q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X a ≤ y Λ f ( a ) χ ( a ) a s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt | t | ≤ X qL
Therefore, our original a -integral is ≪ X qL c > t -integral is ≪ Ψ( x, y ) ϕ ( q )(log x ) A − , which is acceptable. OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 35 Consequences of (5.5) . If χ T q then, by (5.5), we have X x/y 0, we have, using (5.4), X x/y Combining theabove we have therefore proved that1 ϕ ( q ) X χ (mod q ) χ T q ( A,x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x We take T = ϕ ( q ) in the same ar-gument, and extend the range in section 5.5 to | t | ≤ T , getting ridof the need for the final range. In the calculations in that sectionwe replace U by U + 1 in our bounds, and end up with a contri-bution ≪ V Ψ( x, y )(log x ) ε /ϕ ( q ). With our new choice of T , thecontribution with | t | > T also is bounded by the same quantity, so,going through the same process, we end up with the upper bound ≪ V Ψ( x, y )(log x ) ε /ϕ ( q ) ≪ Ψ( x, y )(log x ) ε /ϕ ( q ) on our sum, asdesired.5.10. Extension to f supported on any primes ≤ y . Given f supported on primes p ≤ y , define the multiplicative function g with g ( p k ) = 0 if p ≤ qL , and g ( p k ) = f ( p k ) if p > qL . Thereforeif ψ (mod r ) is a primitive character that induces χ (mod q ) then g ( m ) χ ( m ) = f ( m ) ψ ( m )1 ( m,P )=1 where P = Q p ≤ qL p . Let h ( . ) bethe multiplicative function which is supported only on the prime pow-ers p k , for which p ≤ qL but does not divide r , and defined so that( h ∗ f ψ )( p k ) = 0 if k ≥ 1. Thus h ∗ f ψ = gχ and so(5.8) S g ( x, χ ) = X m ≤ x h ( m ) S f ( x/m, ψ ) . We note that hψ ∈ C as f ∈ C , so that each | h ( m ) | ≤ Deduction of Proposition 5.1 from Proposition 5.2. Assume that ψ isa primitive character mod r with r ≤ R , such that | S f ( X, ψ ) | ≤ Ψ( X, y ) /V (log x ) C for all X in the range x/z < X ≤ x where z = z ( qL ) log log x and z = exp((log log x ) ). Now let x/z ≤ X ≤ x . Then | S f ( X/m, ψ ) | ≤ Ψ( X/m, y ) /V (log x ) C for m ≤ ( qL ) log log x and so, by (5.8) and (2.4),we obtain | S g ( X, χ ) | ≪ Ψ( X, y ) V (log x ) C X m ≤ ( qL ) log log x P ( m ) ≤ qL m α + Ψ( X, y ) X ( qL ) log log x 6∈ T q,g ( A, z ).Proposition 5.2 then implies that, for z = y ( q (log x ) A ) /α ,(5.10) X χ (mod q ) χ ∗ q ( A ) | S g ( x, χ ) | ≪ Ψ( x, y )(log x ) A . OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 39 Write each n ≤ x as n = n ′ N with P ( n ′ ) ≤ qL , and p | N = ⇒ p >qL , so that f ( n ) = f ( n ′ ) g ( N ), and therefore (after renaming n ′ by n ) S f ( x, χ ) = X n ≥ P ( n ) ≤ qL f ( n ) S g ( x/n, χ ) . Now assume that ψ Ξ( C ), so that | S f ( X, ψ ) | ≤ Ψ( X, y ) /V (log x ) C for all x/z < X ≤ x where z = z ( qL ) log log x . Assuming that x/z ≥ x η , (5.10) holds with x replaced by X , for every X in therange x/ ( qL ) log log x ≤ X ≤ x . Then we have, by (5.10), Corollary 5.3and (2.4), X χ (mod q ) χ Ξ q ( C ) | S f ( x, χ ) | ≤ X n ≤ ( qL ) log log x P ( n ) ≤ qL X χ (mod q ) χ Ξ q ( C ) | S g ( x/n, χ ) | + X n> ( qL ) log log x P ( n ) ≤ qL X χ (mod q ) | S g ( x/n, χ ) |≪ Ψ( x, y )(log x ) A X n ≤ ( qL ) log log x P ( n ) ≤ qL n α + Ψ( x, y )(log x ) ε X n> ( qL ) log log x P ( n ) ≤ qL n α ≪ Ψ( x, y )(log x ) A + o (1) , proceeding for these two sums just as we did above.The claimed estimate then follows by taking A = B + ε and so C = 2 B + 2 + 4 ε . To guarantee that x/z ≥ x η we need that y ( q (log x ) A ) /α ( qL ) log log x ≤ x (1 − η ) / , and we already have the restrictions y ≥ V q (log x ) A − and (5.9).These all follow from the bounds on q and y given in the hypothesis.All that remains now is to bound | Ξ | . (cid:3) Bounding the size of Ξ( C ) . The large sieve gives that X r ≤√ X X ψ primitive mod r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m ≤ X f ( m ) ψ ( m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ X Ψ( X, y ) . Therefore, for a given X in the range x η ≤ X ≤ x , the number of excep-tional ψ with r ≤ R is ≪ ( X/ Ψ( X, y ))(log x ) C u O (1) ≪ u u + o ( u ) (log x ) C .If (5.1) holds for X , and | X ′ − X | ≪ X/V (log x ) C then (5.1) holdsfor X ′ . So to obtain the full range for X we sample at X -values spacedby gaps of length ≫ X/V (log x ) C . Therefore the number of excep-tional ψ with r ≤ R in our range is ≪ u u + o ( u ) (log x ) C +1 , as claimed. Deduction of several Theorems Proof of two Theorems on f in arithmetic progressions. Proof of Theorem 1.2 . Let J = k +1 so that ≤ B = 1 − ε < − / √ J in Corollary 3.3. Let w = (log x ) B and exp( C (log log x ) ) < Q ≤ x / − δ , with C > B . Let Ξ = { ψ , . . . , ψ J − } so that ∆ k = ∆ Ξ and(4.1) holds. Then, by Corollary 4.1 and Corollary 3.3 we deduce that X q ∼ Q max ( a,q )=1 | ∆ Ξ ( f, x ; q, a ) | ≪ x (log x ) B . (cid:3) Deduction of Corollary 1.1. We apply Proposition 3.4 whose hypothe-ses are satisfied using Theorem 1.2 for any C > 0, suitably adjustingthe value of ε . (cid:3) On average, supported only on the smooths. Proof of Theorem 1.6. We proceed as in the proof of Theorem 4.2,but now with the parameters Y = exp((log log x ) ) , Z = y and w =(log x ) A . Since Q ≤ x / /y / (log x ) A , Lemma 4.5 implies that E bilinear ≪ x (cid:18) w log w ) / + Y . + (log x ) / π ( Y ) / (cid:19) + QZ / x / ≪ x (log x ) A . Moreover, we have TY log Y ≪ x (log x ) A . The key difference here is in the E sieve term. Since f is only supportedon the y -smooth integers, we now have E sieve ≤ X n ≤ x | F ( n ) | ( n, Q p>Y p )=1 ≤ X q ∼ Q (cid:18) Ψ( x, Y, a q , q ) + 1 q r Ψ( x, Y, a q , q s ) (cid:19) . By (2.6), followed by (2.7), this quantity is ≪ X q ∼ Q q Ψ( x, Y ) + x (log x ) A ≪ x (log x ) A . Thus from Proposition 4.3 we deduce the following variant on Theorem4.2: X q ∼ Q max ( a,q )=1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ xn ≡ a (mod q ) f ( n ) − q r X n ≤ xn ≡ a (mod q s ) f ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ x (log x ) A . OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 41 We will now apply Proposition 1.5 to the sums of f ( n ) over n inthe w -smooth arithmetic progressions a (mod q s ). Therefore the onlyrelevant χ ∈ Ξ will be those with w -smooth conductors. We select B so that 6 B + 7 < A , which implies that (4.1) holds, provided x / − ε ≥ Q ≥ x ε/ (2 log log x ) . By Corollary 4.1 suitably amended withthis input, we then deduce that(6.1) X q ∼ Q max ( a,q )=1 | ∆ Ξ ( f, x ; q, a ) | ≤ w X q s ≤ QP ( q s ) ≤ w max ( a,q s )=1 | ∆ Ξ ( f, x ; q s , a ) | + O (cid:18) x (log x ) A (cid:19) . By Proposition 1.5 we have, for R = x ε/ (3 log log x ) (6.2)1log w X q s ≤ RP ( q s ) ≤ w max ( a,q s )=1 | ∆ Ξ ( f, x ; q s , a ) | ≪ w X q s ≤ RP ( q s ) ≤ w ϕ ( q s ) x (log x ) B ≪ x (log x ) B . For the remaining q s we use the trivial upper bound | ∆ Ξ ( f, x ; q, a ) | ≪ (1 + | Ξ q | ) x/q , to obtain1log w X R ≤ q s ≤ QP ( q s ) ≤ w max ( a,q s )=1 | ∆ Ξ ( f, x ; q s , a ) | ≪ x log w X R ≤ q ≤ QP ( q ) ≤ w | Ξ q | q ≤ x log w X χ ∈ Ξ r χ ≤ Y r χ X q ≥ R/YP ( q ) ≤ w q + X χ ∈ Ξ r χ >Y r χ X q ≥ P ( q ) ≤ w q ≪ x (log x ) A , using (4.1), and that P χ ∈ Ξ , r χ >Y /r χ ≤ | Ξ | /Y , and our estimates onsmooth numbers (as R/Y > w log log x ). We therefore deduce (1.6). (cid:3) Deduction of Corollary 1.7. We apply Proposition 3.4 whose hypothe-ses are satisfied using Theorem 1.6 for any C > B + 7. (cid:3) Breaking the x / -barrier To break the x / -barrier, we need to reduce the Q in the upperbound in Lemma 4.5. This term arises in the estimates that the num-ber of terms in various arithmetic progressions is the length of thatprogression plus O (1). Following ideas from [2, 15, 34] and others, wewill be more precise about all those “ O (1)”s by using Fourier analysisto obtain some cancellationWe will be able to do this when the residue classes a q do not varywith q : Let 1 ≤ | a | ≪ Q ≤ x − δ for some small fixed δ > 0, and considerthe residue classes a (mod q ) when q ∼ Q and ( a, q ) = 1. Let F bethe function defined by F ( n ) = X q ∼ Q ( a,q )=1 ξ q (cid:18) n ≡ a (mod q ) − q r n ≡ a (mod q s ) (cid:19) when n = a , and set F ( a ) = 0. Writing τ for the divisor function, wehave | F ( n ) | ≪ τ ( n − a ) ≤ x o (1) for each n = a and n ≤ x , and so k F k ∞ ≤ x o (1) . Moreover we have the L -bound: X n ≤ x | F ( n ) | ≪ X n ≤ x τ ( n ) ≪ x (log x ) . Proposition 7.1 (Bilinear term beyond the x / -barrier) . Let F bedefined as above, and fix δ > . If ≤ | a | ≪ Q ≤ x − δ and P ≤ x ε for some very small ε > (depending on δ ) then E p,p ′ ∼ P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X m ≤ min( x/p,x/p ′ ) F ( pm ) F ( p ′ m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ xP (cid:18) w log w + P . + (log x ) π ( P ) (cid:19) . We begin by noting that the contribution from those terms with p = p ′ is bounded by1 π ( P ) X p ∼ P X m ≤ x/p | F ( pm ) | ≪ π ( P ) X p ∼ P X n ≤ xn ≡− a (mod p ) τ ( n ) as | F ( pm ) | ≪ τ pm − a . Since τ ( n ) is a positive valued multiplicativefunction, this sum is easily bounded by Shiu’s Theorem, [30], to be ≪ xP · (log x ) π ( P ) , which is acceptable. Now fix p = p ′ and analyze the inner sum over m .We insert a smooth weight into the sum over m : Let η > ψ : R → [0 , 1] be a smooth approxi-mation to the indicator function [0 , with the following properties:(1) R ψ = 1;(2) ψ is supported on [ − η, η ], and ψ = 1 on [ η, − η ];(3) k ψ ( A ) k ≪ A η − A +1 for all A ≥ b ψ decaysrapidly: | b ψ ( h ) | ≪ A (1 + | h | ) − A η − A +1 OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 43 for all A ≥ 1. We may replace the sum over m by the smoothed versionΣ( p, p ′ ) := X m ∈ Z ψ ( m/M ) F ( pm ) F ( p ′ m ) , where M = min( x/p, x/p ′ ), at the cost of an error ≪ ηM k F k ∞ ≪ ηx o (1) · xP . Let 0 < σ ≤ min { δ/ , / } and assume ε = σ/ 3. If we choose η = x − σ , then the error is acceptable since P ≤ x ε . By the definition of F ,we haveΣ( p, p ′ ) = X q,q ′ ∼ Q ( a,q )=( a,q ′ )=1 ξ q ξ q ′ X m ∈ Z ψ ( m/M ) K ( p, p ′ , q, q ′ ; m ) , where K ( p, p ′ , q, q ′ ; m ) is the expression (cid:18) pm ≡ a (mod q ) − q r pm ≡ a (mod q s ) (cid:19) (cid:18) p ′ m ≡ a (mod q ′ ) − q ′ r p ′ m ≡ a (mod q ′ s ) (cid:19) . The inner sum over m can be written as a sum of four terms, the firstof which is X m ∈ Z ψ ( m/M ) · pm ≡ a (mod q ) · p ′ m ≡ a (mod q ′ ) . The sum is empty unless p ′ a ≡ pa (mod ( q, q ′ )) ⇔ p ′ ≡ p (mod ( q, q ′ )) . When this holds, the sum should have a main term of M/ [ q, q ′ ] and anerror term of g ( q, q ′ ) = X m ∈ Z ψ ( m/M ) · pm ≡ a (mod q ) · p ′ m ≡ a (mod q ′ ) − M [ q, q ′ ] . After summing over p, p ′ , the contributions from the four main termslead to exactly Σ + Σ , which were treated in the proof of Lemma 4.5:Σ + Σ ≪ xP (cid:18) w log w + P . + log xπ ( P ) (cid:19) . It suffices to control the error terms by showing that E := X q,q ′ ∼ Q ( a,q )=( a,q ′ )=1 ξ q ξ q ′ g ( q, q ′ ) ≪ x/P ,E := X q,q ′ ∼ Q ( a,q )=( a,q ′ )=1 ξ q ξ q ′ q − r g ( q s , q ′ ) ≪ x/P , E ′ := X q,q ′ ∼ Q ( a,q )=( a,q ′ )=1 ξ q ξ q ′ q ′− r g ( q, q ′ s ) ≪ x/P ,E := X q,q ′ ∼ Q ( a,q )=( a,q ′ )=1 ξ q ξ q ′ ( q r q ′ r ) − g ( q s , q ′ s ) ≪ x/P , for any fixed p = p ′ . By symmetry, it suffices to prove the bounds for E , E , E . We start by analyzing g ( q, q ′ ) for fixed q, q ′ . Lemma 7.2. Suppose that ≤ | a | ≪ Q ≤ x and P ≤ x ε . Fix q, q ′ ≤ Q and p, p ′ ∼ P with d = ( q, q ′ ) satisfying p ≡ p ′ (mod d ) and ( a, qq ′ ) = 1 . Write q = dℓ and q ′ = dℓ ′ so that ( ℓ, ℓ ′ ) = 1 . Then g ( q, q ′ ) = Mq ′ X < | h |≤ H e p ′ ℓ ( kh · pℓ ′ ) Z | u |≤ d/Q ψ ( ℓu ) e dℓ ′ ( − M hu )d u + O σ ( x − / ) , where H = x σ Q / ( dM ) and k = ( p ′ − p ) a/d , assuming < σ ≤ / and ε = σ/ . Here, and in the sequel, the notation pℓ ′ denotes the multiplicativeinverse of pℓ ′ modulo p ′ ℓ , when it appears inside e p ′ ℓ ( · ). The definitionof g ( q, q ′ ) involves [ q, q ′ ] = dℓℓ ′ . A key advantage of Lemma 7.2 is thatthe two variables ℓ and ℓ ′ are separated apart from a term of the forme ℓ ( ∗ ℓ ′ ). Proof. Let r be the unique solution modulo [ q, q ′ ] to the simultaneouscongruence conditions pr ≡ a (mod q ) , p ′ r ≡ a (mod q ′ ) . Then g ( q, q ′ ) = X m ∈ Z ψ (cid:16) mM (cid:17) m ≡ r (mod [ q,q ′ ]) − M [ q, q ′ ] . By the definition of the Fourier transform b ψ and a change of variableswe have b ψ (cid:18) πM h [ q, q ′ ] (cid:19) = b ψ (cid:18) πM hdℓℓ ′ (cid:19) = ℓ Z | u |≤ /ℓ ψ ( ℓu )e (cid:18) − M hudℓ ′ (cid:19) d u, and by Poisson summation, we obtain g ( q, q ′ ) = M [ q, q ′ ] X h ∈ Z \{ } b ψ (cid:18) πM h [ q, q ′ ] (cid:19) e (cid:18) rh [ q, q ′ ] (cid:19) . OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 45 Using the rapid decay of b ψ , the contribution to g ( q, q ′ ) from those termswith | h | ≥ H is ≪ A (cid:14) (cid:18) ηM H [ q, q ′ ] (cid:19) A − ≪ σ x − by the choices of η = x − σ and H and letting A = σ − + 1. We willprove that(7.1) e (cid:18) rh [ q, q ′ ] (cid:19) = e dℓℓ ′ ( rh ) = e p ′ ℓ ( kh · pℓ ′ ) + O (cid:18) | h | Q (cid:19) . The total contribution of these error terms, over all h with 1 ≤ | h | ≤ H ,is therefore, ≪ Q (cid:30) (cid:18) ηM [ q, q ′ ] (cid:19) ≪ Q (cid:18) Q Pηx (cid:19) ≪ x − ( x + σ + ε ) ≪ x − / , as σ ≤ / 75 and ε = σ/ 3, which implies the result when we insert theformulas and estimates above into the sum for g ( q, q ′ ).To establish (7.1), let c (mod d ) be the residue class with pc ≡ p ′ c ≡ a (mod d ), so that r ≡ c (mod d ). Make a change of variables r = ds + c , so thate dℓℓ ′ ( rh ) = e ℓℓ ′ ( sh )e dℓℓ ′ ( ch ) = e ℓℓ ′ ( sh ) + O ( ch/dℓℓ ′ ) . The value of s is determined by the congruence conditions p ( ds + c ) ≡ a (mod q ) , p ′ ( ds + c ) ≡ a (mod q ′ ) , which can be rewritten as ps ≡ b (mod ℓ ) , p ′ s ≡ b ′ (mod ℓ ′ ) , where b = ( a − pc ) /d and b ′ = ( a − p ′ c ) /d . Since ℓ and ℓ ′ are coprime,the Chinese remainder theorem leads to s ≡ b (( pℓ ′ ) − mod ℓ ) · ℓ ′ + b ′ (( p ′ ℓ ) − mod ℓ ′ ) · ℓ (mod ℓℓ ′ ) . Hence e ℓℓ ′ ( sh ) = e ℓ ( bh · pℓ ′ )e ℓ ′ ( b ′ h · p ′ ℓ ) . Now apply the reciprocity relation v − (mod u ) u + u − (mod v ) v ≡ uv (mod 1)with u = pℓ ′ and v = p ′ ℓ to obtaine pℓ ′ ( wp ′ ℓ )e p ′ ℓ ( wpℓ ′ ) = e pp ′ ℓℓ ′ ( w )for any w . This implies thate ℓ ′ ( b ′ h · p ′ ℓ ) = e pℓ ′ ( pb ′ h · p ′ ℓ ) = e p ′ ℓ ( − b ′ h · ℓ ′ )e p ′ ℓℓ ′ ( b ′ h ) = e p ′ ℓ ( − b ′ h · ℓ ′ )+ O ( b ′ h/p ′ ℓℓ ′ ) . The main term is thereforee ℓ ( bh · pℓ ′ )e p ′ ℓ ( − b ′ h · ℓ ′ ) = e p ′ ℓ ( kh · pℓ ′ ) , as claimed. For the error terms we note that | c | ≤ d and | b ′ | ≤ | a/d | + p ′ ;and therefore the error is ≪ (cid:18) | a | dP (cid:19) | h | ℓℓ ′ ≪ (cid:0) P + | a | (cid:1) | h | Q ≪ | h | Q as | a | , x σ ≤ Q and | d | ≤ P ≤ x σ , as claimed. This completes theproof. (cid:3) Now we perform the summation over q, q ′ : Lemma 7.3. Suppose that ≤ | a | ≪ Q ≤ x − δ and P ≤ x ε . For anysequences { γ ( q ) } , { γ ′ ( q ) } with | γ ( q ) | , | γ ′ ( q ) | ≤ we have X q,q ′ ∼ Q ( q,q ′ ) | p − p ′ γ ( q ) γ ′ ( q ′ ) g ( q, q ′ ) ≪ x σ + ε Q − + Q x − / ≪ xP , assuming σ = min { δ/ , / } and ε = σ/ .Proof. The total contribution of the error term from Lemma 7.2 is O ( Q x − / ). For each integer h, ≤ | h | ≤ H := x σ Q /dM , eachdivisor d of p ′ − p , and each fixed | u | ≤ d/Q , we restrict our attentionto those pairs with ( q, q ′ ) = d . Write q = dℓ and q ′ = dℓ ′ , and thendefine α ( n ) = ( γ ( ℓd ) ψ ( ℓu ) if n = p ′ ℓ for some ℓ ∼ Q/d , so that | α ( n ) | ≤ | ψ ( ℓu ) | ≪ 1, and β ( n ′ ) = ( ( Q/ℓ ′ d ) γ ′ ( ℓ ′ d )e ℓ ′ d ( − M hu ) if n ′ = pℓ ′ for some ℓ ′ ∼ Q/d . Theorem 1 of [1] (with A = 1 , θ = kh and the roles of α and β swapped) implies that X n ∼ N, n ′ ∼ N ′ ( n,n ′ )=1 α ( n ) β ( n ′ )e n ( kh · n ′ ) ≪ Qd (cid:18) P Qd (cid:19) + ε where N = p ′ Q/d and N ′ = pQ/d , and k = ( p ′ − p ) a/d , as | kh | / ( P Q/d ) ≪| a | /x − σ ≪ 1. Summing over u and h , our sum is therefore ≪ MQ X d | p − p ′ dQ · x σ Q dM · Qd (cid:18) P Qd (cid:19) + ε ≪ Qx σ ( P Q ) + ε ≪ x σ + ε Q − . OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 47 (cid:3) Completion of the proof of Proposition 7.1. Each of E , E , E can beeffectively bounded using Lemma 7.3. Indeed, for E we apply Lemma 7.3with γ ( q ) = ( ξ q if q ∼ Q γ ′ ( q ′ ) = ( ξ q ′ if q ′ ∼ Q E we apply Lemma 7.3 with γ ( q s ) = X q r ∼ Q/q s ξ q r q s q − r , γ ′ ( q ′ ) = ( ξ q ′ if q ′ ∼ Q E . (cid:3) Proof of Theorems 1.8 and 1.9. For a fixed with 1 ≤ | a | ≪ Q ≤ x − δ , we replace Lemma 4.5 by Proposition 7.1 to obtain ana-logues of Theorem 4.2 and Corollary 4.1. The proof of Theorems 1.8and 1.9 then follows in exactly the same way as the arguments in sec-tion 6. 8. Various examples The large prime obstruction; the proof of Proposition 1.3. In this section we will construct multiplicative functions with certainproperties. We will only define these multiplicative functions on [1 , x ]since the integers n > x do not contribute. Let P be the set of primes p ∈ ( x/ , x ] for which there does not exist an integer q ∼ Q with q | p − f + , f − be the multiplicative functions for which f + ( n ) = f − ( n ) = g ( n ) for all n ≤ x with n 6∈ P , and f + ( p ) = 1 and f − ( p ) = − p ∈ P . If f ∈ { f + , f − } then X n ≤ xn ≡ a (mod q ) f ( n ) = X n ≤ xn ≡ a (mod q ) g ( n ) + X x/
1) = X x/
6∈ P , the first term on the right-hand side of thelast displayed equation vanishes, by the definition of P . Therefore∆( f + , x ; q, − ∆( f − , x ; q, 1) = 2 P ϕ ( q ) ∼ ϕ ( q ) x log x . The asymptotic in the last step follows from Theorem 6 and Corollary2 of Ford’s masterpiece [14] which imply that almost all primes p inthe interval x/ < p ≤ x belong to P . Thus for each q we haveeither | ∆( f + , x ; q, | or | ∆( f − , x ; q, | is ≫ π ( x ) /ϕ ( q ). The conclusionfollows immediately.8.2. f satisfying the Siegel-Walfisz criterion but not Bombieri-Vinogradov with prime moduli. We will assume (1.5) in what fol-lows. Let x / < Q < x / − ε , and let P now be the set of primes p ∈ ( x/ , x ] for which there exists a prime q ∼ Q for which q | p − 1. Wenow define the multiplicative function f so that f ( p ) = − p ∈ P ,and f ( n ) = 1 if n ≤ x and n / ∈ P . Therefore if m < x/ a, m ) = 1then∆( f, x ; m, a ) = ∆(1 , x ; m, a ) − P , x ; m, a ) = − P , x ; m, a )+ O (1) . Note that if p ∈ P then p − q ∼ Q . Therefore, by inclusion-exclusion we have P = X q ∼ Qq prime π ∗ ( x ; q, − X q 0, for the sums OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 49 over n ≤ x . The same argument works for sums over n ≤ X , for any X ≤ x , changing the definition of π ∗ ( X ; q, a ) to be 0 if X ≤ x/ 2, and π ∗ ( X ; q, a ) := π ( X ; q, a ) − π ( x/ q, a ) if x/ < X ≤ x .On the other hand, (1.5) (or the unconditional Brun-TitchmarshTheorem), implies that P ≪ X q ∼ Qq prime π ∗ ( x ) ϕ ( q ) ≪ x (log x ) . Therefore if q ∼ Q and q is prime then∆(1 P , x ; q, 1) = π ∗ ( x ; q, − P ϕ ( q ) ∼ π ∗ ( x ) ϕ ( q ) . Therefore if q ∼ Q and q is prime then∆( f, x ; q, ∼ − q · x log x , and so X q ∼ Q | ∆( f, x ; q, | ≥ X q ∼ Qq prime | ∆( f, x ; q, | ≫ x (log x ) . This doesn’t quite prove Corollary 1.4, but it will be fixed in the nextsubsection.8.3. f satisfying the Siegel-Walfisz criterion but not Bombieri-Vinogradov; the proof of Corollary 1.4. Now we use a minormodification of the argument in the previous subsection to extend theestimate to all moduli. Let x / < Q < x/ 2, and I = ( x / , x / ].Let P now be the set of primes p ∈ ( x/ , x ] for which there exists aprime ℓ ∈ I with ℓ | p − 1. Define the multiplicative function f so that f ( p ) = − p ∈ P , and f ( n ) = 1 if n ≤ x and n / ∈ P . So we againhave that if m < x/ a, m ) = 1 then∆( f, x ; m, a ) = − P , x ; m, a ) + O (1) . The argument of Section 8.2 yields the Siegel-Walfisz criterion for f ,assuming (1.5). We also have, assuming (1.5), that P ≤ X ℓ ∈ Iℓ prime π ∗ ( x ; ℓ, 1) = { o (1) } X ℓ ∈ Iℓ prime π ∗ ( x ) ℓ − { c + o (1) } π ∗ ( x ) , where c = log(6 / < / Now suppose that q ∼ Q (not necessarily prime) and that thereexists a prime ℓ in the interval I which divides q . If x/ < p ≤ x and p ≡ q ) then ℓ | q | p − p ∈ P . So we have∆(1 P , x ; q, 1) = π ∗ ( x ; q, − P ϕ ( q ) ≥ (1 − c + o (1)) π ∗ ( x ) ϕ ( q ) ≥ · π ∗ ( x ) ϕ ( q ) , and therefore − ∆( f, x ; q, ≫ ϕ ( q ) · x log x . Now each such q can have at most one such prime divisor ℓ . Therefore X q ∈ Q | ∆( f, x ; q, | ≫ x log x X ℓ ∈ Iℓ prime X q ∼ Qℓ | q ϕ ( q ) ≫ x log x X ℓ ∈ Iℓ prime ℓ ≫ x log x . This completes the proof of Corollary 1.4.8.4. f that correlate to many characters. For a given integer q suppose we are given constants g ( χ ) for each character χ (mod q ), anddefine the multiplicative function f byΛ f ( n ) = c q ( n )Λ( n ) , where c q ( n ) := X χ (mod q ) g ( χ ) χ ( n )depends only on n (mod q ). This implies that the Dirichlet series F associated to f is F ( s ) = Y χ (mod q ) L ( s, χ ) g ( χ ) . Moreover if we twist f by a character ψ (mod q ), then F ψ ( s ) := X n ≥ ( f ψ )( n ) n s = Y χ (mod q ) L ( s, χ ) g ( χψ ) . By the Selberg-Delange Theorem (see, e.g., Theorem 5 of Section II.5.5in [32]) one has that if g ( ψ ) is not an integer then | S f ( x, ψ ) | ≫ x (log x ) − Re( g ( ψ )) . Now f ∈ C if and only if | c q ( n ) | ≤ n . We will make theparticular choice that g ( χ ) = 1 ϕ ( q ) X a (mod q ) χ ( a ) e ( a/q ) OMBIERI-VINOGRADOV FOR MULTIPLICATIVE FUNCTIONS 51 are Gauss sums (where e ( t ) := e iπt ), so that c q ( n ) = 1 ϕ ( q ) X χ (mod q ) χ ( n ) X a (mod q ) χ ( a ) e ( a/q )= X a (mod q ) e ( a/q ) 1 ϕ ( q ) X χ (mod q ) χ ( n ) χ ( a ) = e ( n/q ) . Therefore f ∈ C and | g ( χ ) | = √ q/ ( q − 1) whenever χ is non-principaland q is prime. Moreover Katz showed (though see [25] for a mucheasier proof) that the arguments of the g ( χ ) are equi-distributed aroundthe unit circle. In particular the number of non-principal ψ (mod q )for which 1 − Re( g ( ψ )) > − √ q is ∼ q/ 3. We deduce that if q is aprime for which 3 k . q < /ǫ then there are > k primitive characters ψ (mod q ) for which | S f ( x, ψ ) | ≫ x (log x ) − ǫ . Therefore, by (3.2) and the prime number theorem, we deduce that forany given integer k there is an ε ′ ∼ √ k , for which there exists f ∈ C such that X q ∼ Q max a : ( a,q )=1 | ∆ k ( f, x ; q, a ) | ≫ x (log x ) − ε ′ . This shows that Theorem 1.2 is close to “best possible”.9. Average of higher U k -norms In this section, we investigate a higher order generalization of theBombieri-Vinogradov inequality, which measures more refined distri-butional properties. This higher order version involves Gowers norms,a central tool in additive combinatorics. We refer the readers to [31,Chapter 11] for the basic definitions and applications. In particular, k f k U k ( Y ) stands for the U k -norm of the function f on the discrete in-terval [0 , Y ] ∩ Z .For any arithmetic function f : Z → C and any residue class a (mod q ), denote by f ( q · + a ) the function m f ( qm + a ). Corollary 9.1. Fix a positive integer k . Fix B ≥ and ε > . Let ≤ y ≤ x / be large. Let f be a completely multiplicative functionwith each | f ( n ) | ≤ , which is only supported on y -smooth integers.Assume that (9.1) X n ≤ xn ≡ a (mod q ) f ( n ) ≪ C x (log x ) C for any ( a, q ) = 1 and any C ≥ . Let Q = x / / ( y / (log x ) A ) , where A is sufficiently large in terms of k, B, ε . Then for all but at most Q (log x ) − B moduli q ≤ Q , we have max ≤ a In view of Corollary 1.7, the hypothesis on f implies that(9.2) X q ≤ Q max ( a,q )=1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ xn ≡ a (mod q ) f ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ x (log x ) B ′ , for any constant B ′ = B ′ ( B ), provided that A is chosen large enough.The conclusion of Corollary 9.1 follows by using this estimate in placeof the Bombieri-Vinogradov inequality for the Liouville function λ inthe argument in [29, Section 2.2].More precisely, if q ≤ Q is an exceptional moduli in the sense that k f ( q · + a q ) k U k ( x/q ) ≥ ε for some residue class a q (mod q ), where 0 ≤ a q < q and ( a q , q ) = 1,then by the inverse theorem for Gowers norms [22] f must correlatewith a nilsequence of complexity O ε (1) on the progression { n ≤ x : n ≡ a q (mod q ) } . 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E-mail address : [email protected] Department of Mathematics, University of Kentucky, 757 Patter-son Office Tower, Lexington, KY, 40506, USA E-mail address ::RP ( q ) ≤ w q + x X χ ∈ Ξ r χ X R/r χ