Combinatorial identities and Titchmarsh’s divisor problem for multiplicative functions Sary Drappeau, Berke Topacogullari
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Sary Drappeau, Berke Topacogullari. Combinatorial identities and Titchmarsh’s divisor problem for multiplicative functions. Algebra & Number Theory, Mathematical Sciences Publishers 2019, 13-10, pp.2383-2425. 10.2140/ant.2019.13.2383. hal-01993086
HAL Id: hal-01993086 https://hal.archives-ouvertes.fr/hal-01993086 Submitted on 24 Jan 2019
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. COMBINATORIAL IDENTITIES AND TITCHMARSH’S DIVISOR PROBLEM FOR MULTIPLICATIVE FUNCTIONS
SARY DRAPPEAU AND BERKE TOPACOGULLARI
Abstract. Given a multiplicative function f which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum P f(n)τ(n − h), where τ denotes the |h|
1. Introduction Understanding correlations of arithmetic functions is a fundamental question in analytic number theory. In an explicit form, the problem can be stated as determining the asymptotic behaviour of the sum X (1.1) f(n)g(n − 1), 1 Date: December 17, 2018. 2010 Mathematics Subject Classification. Primary: 11N37; Secondary: 11N25. We thank O. Ramaré, H. L. Montgomery, R. C. Vaughan, R. de la Bretèche, É. Fouvry and G. Tenenbaum for helpful discussions and remarks on the present work. In particular we thank É. Fouvry and G. Tenenbaum for remarks which led to the second proof presented here. Part of this work was done during a visit of BT to Aix-Marseille university, supported by the French-Austrian joint project MuDeRa (FWF I-1751-N26, ANR-14-CE34-0009). 1 2 SARY DRAPPEAU AND BERKE TOPACOGULLARI can be obtained from analogues of the Bombieri-Vinogradov and Brun-Titchmarsh inequalities. We refer to [Gre18, GS18] for recent works on this topic. It is a considerably more difficult problem to obtain full asymptotic expansions for (1.2), say, with an error term of the form O(x(log x)−N ) where N > 0 is fixed but can be chosen arbitrarily large. The gap in difficulty is related to the “x1/2”-barrier for primes in arithmetic progressions on average over moduli. To our knowledge asymptotic expansions are known for only very few specific examples of functions f of arithmetic interest: – the indicator function of primes [Fou85, BFI86], – the indicator function of integers without large prime factors [FT90, Dra15], – the k-fold divisor functions τk(n), k ∈ N, k ≥ 2 [Mot80, Top16, Top17b]. The methods from the last example can also be used to handle the case where f is given by Fourier coefficients of GL2-automorphic forms, although this does not seem to be worked out explicitly in the literature. The purpose of the present paper is to introduce two new methods which lead to an asymptotic expansion for (1.2) for a wide class of multiplicative functions. Let A, D ≥ 1 be fixed integers. Define FD(A) to be the set of all multiplicative functions f : N → C which are D-periodic over the primes in the sense that f(p1) = f(p2) for any primes p1 and p2 with p1 ≡ p2 mod D, and which satisfy the growth condition, |f(n)| ≤ τA(n) for all n ∈ N, where τA(n) denotes the generalized divisor function. Our main result is the following preliminary asymptotic formula for the sum (1.2) for f ∈ FD(A). Theorem 1.1. Let A, D, N ≥ 1. For all f ∈ FD(A) and all x ≥ 2, we have X X X 1 X x (1.3) f(n)τ(n − 1) = 2 f(n)χ(n) + O N , √ ϕ(q) (log x) 1 Theorem 1.2 (General shifts). Let A, D, N ≥ 1. There exists an absolute constant δ > 0, such that, δ for all f ∈ FD(A), all x ≥ 2 and all a, h ∈ Z satisfying 1 ≤ a, |h| ≤ x , we have X x f(n)τ(an − h) = M (x; h, a) + O τ((a, h)) , f (log x)N |h|/a where Mf (x; a, h) is given by χ h X X (h,q) X an Mf (x; a, h) := 2 f(n)χ (an,q) , √ q χ primitive q≤ ax ϕ (h,q) q2/a≤n≤x cond(χ)|D q cond(χ)| (q,h) (an,q)=(h,q) and where the implied constant depends only on A, D and N. Unfortunately, the range of uniformity in h in Theorem 1.2 is comparatively short. This is due to a known uniformity issue of arguments based on exponential sums estimates underlying our bilinear sums estimate (see [FI83, p. 200]). Out of the same reason, the methods used here are not able to address the dual problem N−1 X f(n)τ(N − n) n=1 (for which results are available for instance when f = τ or f = τ3, see [Mot94, Top16]). We mention that results are known for affine correlations whose linear parts are pairwise indepen- dent [Mat12, Mat16], or when there is an additional, long enough average over the shift [Mik92, MRT17a, MRT17b]. See also [ABSR15, BSF17] for a function field analogue in the large q limit. Finally, we mention the work of Pitt [Pit13]. He considered an analogue of the Titchmarsh divisor problem (see Section 1.3) with the divisor function replaced by Fourier coefficients of holomorphic cusp forms. In many situations, these Fourier coefficients and the divisor function exhibit a similar behaviour, since the latter can also be viewed as the Fourier coefficients of an Eisenstein series (see e.g. [Iwa02, Chapter 3.4]). Remarkably, Pitt obtained an estimate with a power saving in the error term unconditionally, something which is not known for the original Titchmarsh divisor problem. It seems possible that his ideas can be adapted to our setting, and that one might obtain an analogue of Theorem 1.2 with the divisor function replaced by Fourier coefficients of holomorphic cusp forms and with a power saving in the error term. We do not pursue this here. We apply Theorem 1.2 to three functions f of particular arithmetic interest: (1) the generalized divisor functions τz(n) with z ∈ C, (2) the indicator function of integers n which are norms of an integral ideal in an abelian extension, (3) the indicator function of integers n with exactly k different prime factors. 1.1. Correlations of divisor functions. Our first application is related to the generalized additive divisor problem, which asks for an asymptotic evaluation of X Dk,`(x, h) := τk(n)τ`(n + h) |h| where Pk,h is a degree k polynomial depending on h, with 2 2 θ2 = 3 and η2 = 3 [DI82a, Mot94], 8 2 θ3 = 9 and η3 = 3 [FI85, Top16], 4 56 15 θk = max 1 − 15k−9 , 57 and ηk = 19 (k ≥ 4 fixed) [Lin63, FT85, Top17b]. 4 SARY DRAPPEAU AND BERKE TOPACOGULLARI In the case k = ` = 2, a similar asymptotic formula holds in a much larger range of uniformity for h, although with a weaker error term (see [Meu01] for the currently best results in this direction). For k, ` ≥ 3 the problem remains completely open. The functions τk are special cases of coefficients of the Dirichlet series ∞ X τz(n) := ζ(s)z for z ∈ and Re(s) > 1. ns C n=1 On prime powers, they are given explicitly by z + ` − 1 (1.4) τ (p`) = . z ` The functions τz for z 6∈ N have a more complicated behaviour than those for z ∈ N. When z = −1 for instance, we recover the Möbius function τ−1(n) = µ(n). Theorem 1.2 leads to an asymptotic expansion of Dz,2(x, h) for arbitrary z ∈ C, uniformly in any fixed disk |z| 1. Theorem 1.3. Let A, N ≥ 1 and ε > 0. There exist a constant δ > 0 and holomorphic functions λh,` : C → C, such that, for |z| ≤ A, x ≥ 2 and 1 ≤ |h| ≤ xδ, N Re(z) X X λh,`(z) x(log x) (1.5) τ (n)τ(n + h) = x(log x)z + O , z (log x)` (log x)N+1−ε |h| The coefficients λh,`(z) can be computed explicitly; see (8.4) infra for an expression of the leading coefficient. If z is a non-positive integer, all the coefficients λh,`(z) vanish and (1.5) effectively becomes an upper bound. Our method leads to a power saving error term in Theorem 1.3 when z = k ∈ N. This is solely due to the fact that in these cases the k-th power of Dirichlet L-functions L(s, χ)k can be continued analytically to a strip Re(s) ≥ 1 − δ for some δ > 0 (excluding the possible pole at s = 1). We do not focus of the case z ∈ N here, since the works mentioned above then give quantitatively stronger estimates. 1.2. Norms of integral ideals. Let K/Q be a Galois extension with discriminant ∆K . We define NK := {N(α): α ideal of OK , α 6= 0}. This set has a rich multiplicative structure, described by the Artin reciprocity law. When the extension is abelian, the Dedekind function ζK (s) factorizes into Dirichlet L-functions mod ∆K , so that the integers in NK can be detected by looking at the congruence classes of their prime factors mod ∆K . Theorem 1.2 eventually applies and leads to the following result. Theorem 1.4. Let K/Q be an abelian field extension. Let N ≥ 1 and ε > 0. There exist a constant δ > 0 δ and real numbers κh,`(K), such that, for x ≥ 2 and 1 ≤ |h| ≤ x , N X X κh,`(K) x (1.6) τ(n − h) = x(log x)1−1/[K:Q] + O , (log x)` (log x)N+1/[K:Q]−ε |h| An interesting special case is given by the extension Q(i)/Q. In this case, NQ(i) is simply the set of integers which can be written as a sum of two squares, and Theorem 1.4 takes the following form. Corollary 1.5. Let B be the set of all integers which can be written as a sum of two squares. Let N ≥ 1 δ and ε > 0. There exist a constant δ > 0 and real numbers βh,`, such that, for x ≥ 2 and 1 ≤ |h| ≤ x , N X 1 X βh,` x (1.7) τ(n − h) = x(log x) 2 + O . (log x)` (log x)N+1/2−ε |h| The first term in the asymptotic formula for the left-hand side of (1.7) can also be obtained using a recent extension of the Bombieri-Vinogradov theorem due to Granville and Shao [GS18], along with the Brun-Titchmarsh inequality. The coefficients κh,`(K) and βh,` can be computed explicitly; see (8.5) infra for an evaluation of the leading coefficient βh,0 in (1.7). Note that, since the indicator function b(n) of the set B correlates with both the principal and the non-principal character mod 4, there are two genuine contributions on the right-hand side in (1.3) when f(n) = b(n). This also explains the discrepancy between the conjectures made in [Iwa76] and [FKR17] on autocorrelations of b(n). We stress that the multiplicity of representations as ideal norms in Theorem 1.5 is not taken into account. Thus the estimate (1.7) is more difficult to obtain than an estimate for the correlation sum X 2 2 2 r2(n)τ(n − h) with r2(n) := {(r, s) ∈ Z : r + s = n} , |h| 1.3. Integers with k prime divisors. The Titchmarsh divisor problem, posed in 1930 [Tit30], asks for an asymptotic evaluation of the sum X (1.8) τ(p − h), |h| k Theorem 1.6. Let N ≥ 1 and ε > 0. There exist a constant δ > 0 and polynomials Ph,`(X) of degree k − 1 such that, for 1 ≤ k log log x and |h| ≤ xδ, k k X X Ph,`(log log x) x(log log x) (1.11) τ(n − h) = x + O , (log x)` k!(log x)N+1−ε |h| 1.4. Overview of the proof of Theorem 1.2. For the sake of clear exposition, we will focus here on the case D = 1, as our arguments extend without much difficulty to the case of general moduli and the arising complications are mainly of technical nature. Note that any f ∈ F1(A) can be approximated (in the convolution sense) by a suitable generalized divisor function, so that it suffices to consider the case f = τz with z ∈ C. We will give two distinct proofs of Theorem 1.2. They are based on two different kinds of combinatorial identities for the generalized divisor function τz, both of which we believe are of independent interest. Our first approach relies on an effective combinatorial formula of Heath-Brown’s type for the divisor function τα with α ∈ Q, and an interpolation argument in the z-variable for weighted mean values of τz. Our second apprach, which is more direct and avoids the interpolation step, is instead based on 1 an identity of Linnik type for τz and the well-factorability of friable numbers . 1.4.1. Proof by Heath-Brown’s identity and interpolation. Our first proof of Theorem 1.2 divides into two parts: We first prove the theorem for rational z, and then extend this result to all z ∈ C. For z ∈ Q, the general structure of the proof of Theorem 1.2 follows the setup of [Fou85, BFI86] (see also [Fou84]). The strategy naturally splits into two steps: (1) We decompose the function f into convolutions with either large smooth components (type I) or suitably localized components (type II). (2) We solve the question for both types of sums. The bulk of the present work concerns the first step. Combinatorial decompositions for prime num- bers have a long history since the works of Vinogradov [Vin37] (we refer to the survey [Ram13] for an account and further references). Yet, it was not until recently that analogous identities emerged for generalized divisor functions. Montgomery and Vaughan (private communication) have recently devel- oped a combinatorial identity of Vaughan’s type [Vau75] for τ1/2, which initially motivated largely the present work. Unfortunately, as for primes, the bilinear sums coming from a raw application of this identity are not quite localized enough to be effective for Titchmarsh’s problem, and even though this can sometimes be fixed by iterating the formula [Fou81], our early attempts were unsuccessful. Instead we follow the more flexible approach of Heath-Brown [HB82] (which is related to [Gal68]). Our first result (Theorem 3.2 below) is a uniform combinatorial formula of Heath-Brown’s type for the divisor function τ u with u/v ∈ . In the simplest case 0 < u < v, it reads v Q K X X X (1.12) τ u (n) = c`,K,u/v ··· τ 1 (n1) ··· τ 1 (n`v−u) for n ≤ x, v − v − v `=1 m1···m`n1···n`v−u=n 1/K n1,...,n`v−u≤x where K ∈ N>0 is arbitrary and where c`,K,u/v ∈ Q. A more general formula holds for any rational number u/v (see Theorem 3.2). A crucial property of this formula is that it is sensitive almost only to the archimedean size of u/v. Indeed, for |u/v| ≤ A, the coefficients c`,K,u/v, the length of the `-sum and the value at primes n = p of each `-summand on the right-hand side are bounded in terms of A and K only (but not of v). Thus, the only loss due to the size of v comes from the number O(v) of terms in the convolution, which has essentially no effect on what follows. In the same way, we can express any rational convolution power ∗u/v f of a multiplicative function in terms of higher convolutions ∗k f with 1 ≤ k ≤ K and a bilinear term with one component supported on the interval [xε, x1/K ]. However, to our knowledge asymptotic formulae for the correlation sums X (1.13) (∗kf)(n)τ(n + 1), n≤x for k ≥ 2 are currently known for only very few functions f (essentially constant functions and Dirichlet characters). This is the main obstacle towards using decompositions of this form to prove Theorem 1.2 for complex-fold convolutions of multiplicative functions. Regarding the second step, we are mostly able to use the harmonic analysis arguments underly- ing [Fou85, BFI86]. They are based on bounds on Kloosterman sums on average [DI82b], along with Voronoi summation (for type I) and Linnik’s dispersion method (for type II). We will follow the treat- ment made in [Dra17, Top17b], although some work is needed in order to cast the main terms from these works in a form suitable for us. 1The second proof was found only after a preliminary version of the present manuscript was uploaded online. TITCHMARSH’S PROBLEM FOR MULTIPLICATIVE FUNCTIONS 7 Eventually, the arguments described above yield a proof of Theorem 1.2 for f = τ u uniformly in the v range v ≤ (log x)N . As it turns out, this is already sufficient information to be able to conclude. To see why, we return to the correlation sum X D(z) := τz(n)τ(an − h) |h|/a 7/12+ε for y ≥ x and f ∈ FD(A), as well as theorems of Bombieri-Vinogradov type. However, unlike Titchmarsh’s divisor problem, such results could in principle also be obtained by zero-density estimates for Dirichlet L-functions (see [IK04, Chapter 10.5], [Bom65]). 1.4.2. Proof by Linnik’s identity. Our second proof uses a different decomposition for τz, which has the major advantage that it holds uniformly for all z in a fixed bounded subset of C. This avoids the interpolation step necessairy in the first proof, although the resulting combinatorial identity is not as elegant as the identity of Heath-Brown’s type described above. A naive attempt to find a combinatorial formula for τz which is uniform in z might start with Linnik’s formula [IK04, §13.3], which relies on the Taylor series expansion X z ζ(s)z = (1 + (ζ(s) − 1))z = (ζ(s) − 1)j. j j≥0 The main technical difficulty at this point is to truncate the sum over j. In the context of Linnik’s formula, this truncation is performed by restricting to almost-primes from the outset (or inserting a sieve weight), see [Lin63, p.21], but unfortunately this approach is not available in our situation. Instead we write ζ(s) = ζy(s)My(s), where −1 Y 1 ζ(s) ζy(s) := 1 − s and My(s) := , p ζy(s) p≤y with y = x1/K for some K ∈ N, and then apply the Taylor series expansion only on the second factor My(s), so that X z ζ(s)z = ζ (s)z (M (s) − 1)j. y j y j≥0 This expression has the advantage that the j-th summand has no coefficient for n ≤ yj in its Dirichlet series expansion. After expanding and comparing the Dirichlet coefficients on both sides, we are therefore 8 SARY DRAPPEAU AND BERKE TOPACOGULLARI led to the following “raw” combinatorial decomposition (see Theorem 3.3), X X τz(n) = c` τz−`(n1)τ`(n2) for n ≤ x, 0≤` where the c` are some complex numbers which depend on z, but which can be bound uniformly for z 1 (we recall that an integer is said to be y-friable if all of its prime factors are bounded by y). In order to apply this formula, it is of course necessairy to be able to control the factors τz−`(n1). However, the characteristic function of y-friable numbers has good factorability properties (see [Vau89, p.66] or [FT96, Lemme 3.1]): we can essentially replace them in the formula above by convolutions of sequences supported on [1, y] (see Lemma 3.4). This in turn enables us to apply estimates of type I and type II, leading eventually to the desired asymptotic formula. Plan. In Section 2, we introduce our main notations and the subsets of functions of FD(A) we will mainly work with. In Section 3, we present the combinatorial decompositions for τz, on which our proofs are based. In Section 4, we state some auxiliary computations in order to use the results of [Top17b, Dra17]. In Sections 5 and 6, we proof Theorem 1.2 using the combinatorial identity of Heath-Brown’s type, first by treating the case of rational parameters, and then by interpolating the obtained results to all functions in FD(A). In Section 7, we sketch an alternative proof using the combinatorial identity of Linnik’s type. Finally, in Section 8, we estimate the main terms and prove Theorems 1.3, 1.4 and 1.6. 2. First reductions 2.1. Statement of the main proposition. For n, h ∈ Z with n ≥ 1 and n − h ≥ 1, let X 1 X h n (2.1) τh(n; R) := 2 q χ( (h,q) )χ( (h,q) ). e √ ϕ( ) q≤ n−h (h,q) χ (mod q/(h,q)) (n,q)=(h,q) cond(χ)≤R √ Note that τeh(n; R) = τ(n − h) if R > n − h and n − h is not a perfect square. We will eventually choose R of size (log n)O(1). We have a trivial bound ε 1+ε (2.2) τeh(n; R) ε n R . The function τeh(n; R) should be thought of as an approximation to τ(n − h) on average. The main work in proving Theorem 1.2 consists in showing that, for any f ∈ FD(A), we have X X (2.3) f(n)τ(an − h) ∼ f(n)τeh(an; Rx) for x → ∞, n≤x n≤x where Rx is some slowly growing function in x (some appropriate power of log x). Once this is estab- lished, we can evaluate the sum on the right by standard methods. In view of this, it is convenient to define X ∆h(n; R) := τ(n − h) − τeh(n; R) and Σf (I; a, h; R) := f(n)∆h(an; R), n∈I for any interval I ⊆ R+. The main part of this article is concerned with proving the following propo- sition, which puts the statement (2.3) into precise terms, and from which the results described in the introduction can be deduced easily (see Section 8). Proposition 2.1. Let A, D ≥ 1 be fixed. Then we have, for x ≥ 3, I ⊂ [x/2, x] an interval and f ∈ FD(A), the following estimate, x(log x)B (2.4) |Σ (I; a, h; R)| ≤ Cτ((a, h)) for 1 ≤ a, |h|,R ≤ xδ, f R1/3 where δ > 0 is some absolute constant and where B, C > 0 are constants which depend only on A and D. 2.2. Restricting the set of functions. It is known in multiplicative number theory that, to a certain degree of precision, the magnitude of the mean value of a multiplicative function f depends mostly on the values f(p), p prime. The following lemma quantifies the analogous phenomenon in our case. Lemma 2.2. Let f, g : N → C be multiplicative functions, which satisfy the following conditions, (i) |g(n)| ≤ τM (n) for some M ≥ 1 and all n ∈ N, X (f ∗ g−1)(n) (ii) H := < +∞ for some σ < 1. nσ n≥1 TITCHMARSH’S PROBLEM FOR MULTIPLICATIVE FUNCTIONS 9 Furthermore, assume there are constants %, δ ∈ (0, 1] and B,C ≥ 1 such that, for all x ≥ 1 and all intervals I ⊂ [x/2, x], x(log x)B (2.5) |Σ (I; a, h; R)| ≤ Cτ((a, h)) for 1 ≤ a, |h|,R ≤ xδ. g R% Then there exists C0, δ0 > 0 depending only on %, δ, σ and M, such that, for all x ≥ 1 and all intervals I ⊂ [x/2, x], B x(log x) 0 (2.6) |Σ (I; a, h; R)| ≤ HCC0τ((a, h)) for 1 ≤ a, |h|,R ≤ xδ . f R% Proof. Let h := f ∗ g−1. We have X Σf (I; a, h; R) = g(n1)h(n2)∆h(an1n2; R) n1n2∈I X X = h(n2)Σg(I/n2; an2, h; R) + h(n2)Σg(I/n2; an2, h; R), n2≤T n2>T for some parameter T ≥ 1. For the sum on the left we use the assumption (2.5), so that X x(log x)B h(n )Σ (I/n ; an , h; R) CHτ((a, h)) , 2 g 2 2 σ R% n2≤T provided that the parameters a, h and R satisfy xδ xδ 1 ≤ a ≤ and 1 ≤ |h|,R ≤ . T 1+δ T δ 1+ε For the sum on the right we use the trivial bound Σg(I/n2; an2, h; R) ε,M Rx /n2, and get X 1+ε −1+σ h(n2)Σg(I/n2; an2, h; R) ε,M x RT H. n2>T δ/3 0 δ δ(1−σ) The lemma follows on setting T = x and δ = min( 3 , 4(1+ρ) ). In view of this, in order to prove Proposition 2.1, we will restrict to the following two subsets of FD(A). τ The first subset, denoted by FD(A), consists of functions f : N → C, which are the coefficents of Dirichlet series of the form ∞ X f(n) Y (2.7) = L(s, χ)bχ , ns n=1 χ mod D τ where the parameters bχ are complex numbers such that |bχ| ≤ A. Note that τz ∈ FD(A) for |z| ≤ A.A τ Q τ particularly important role will be played by the subset FD (A) ⊂ FD(A) formed by functions of this form where all the parameters bχ are rational. ω The second subset FD(A) is defined to be the set of functions f : N → C, which are the coefficients of Dirichlet series of the form ∞ X f(n) Y Y zr (2.8) = 1 + , ns ps − 1 × n=1 r∈(Z/DZ) p≡r mod D ω(n) where the coefficients zr are complex numbers such that |zr| ≤ A. This includes the functions n 7→ z for all |z| ≤ A. τ ω Lemma 2.3. For any f ∈ FD(A), there exist g1 ∈ FD(A) and g2 ∈ FD(A) which satisfy the condi- 2 tions (i)–(ii) stated in Lemma 2.2 for σ = 3 , and M,H bounded only in terms of A and D. τ Proof. We first prove the lemma with respect to the set FD(A). Let f ∈ FD(A) be fixed, and let vf : Z → C be the D-periodic function defined by ( f(p) if there exists a prime p such that (p, D) = 1 and p ≡ r mod D, (2.9) vf (r) = 0 if (r, D) > 1. We then set 1 X b := v (r)χ(r) for any character χ mod D, χ ϕ(D) f r (mod D) 10 SARY DRAPPEAU AND BERKE TOPACOGULLARI and define g1 as the coefficients of the following Dirichlet series, ∞ X g1(n) Y (2.10) := L(s, χ)bχ . ns n=1 χ (mod D) −1 We have (f ∗g1 )(p) = 0 if p - D. Moreover, since |bχ| ≤ A, we get |g1(n)| ≤ τAD(n) for all n. Therefore, −1 X (f ∗ g )(n) Y 1 Y 1 1 = 1 + O 1 + O = O (1). n2/3 A,D p2/3 A,D p4/3 A,D n≥1 p|D p-D This proves the first part of the lemma. For the second part, we define g2 by its Dirichlet series ∞ X g2(n) Y Y vf (r) := 1 + . ns ps − 1 × n=1 r∈(Z/DZ) p prime p≡r mod D The fact that g2 satisfies the required conditions can be shown using similar computations as above. Let us at this point also note the following result, which is an easy consequence of the proofs of Lemmas 2.2 and 2.3, and which will become useful later on. Lemma 2.4. Let f ∈ FD(A) and let ψ mod q be a Dirichlet character. Then the Dirichlet series associated to ψf is given by ∞ X ψ(n)f(n) Y = H (s) L(s, ψχ)bχ for Re(s) > 1, ns ψ n=1 χ mod D 1 where Hψ(s) is some holomorphic function defined in Re(s) > 2 and where 1 X b := v (r)χ(r), χ D f r mod D 1 with vf (n) as defined in (2.9). Moreover, for any fixed σ0 > 2 , we have Hψ(s) 1 uniformly in Re(s) > σ0, with the implicit constant depending at most on σ0, A and D. From Lemmas 2.2 and 2.3, we deduce the following statement. Lemma 2.5. To prove Proposition 2.1 in full generality, it suffices to prove it under either one of the τ ω additional hypotheses f ∈ FD(A) or f ∈ FD(A). 3. Combinatorial identites for τz(n) In this section we describe the two combinatorial identites for the generalized divisor function τz on which the proofs of Theorem 1.2 are based. 3.1. A generalization of Heath-Brown’s identity. We first derive a combinatorial decomposition analogous to [HB82] for the function n 7→ τα(n) in the case α ∈ Q. Our argument is based on the following polynomial identity. Lemma 3.1. Let u and v be integers such that v > u ≥ 0. Let K ≥ 1 and N ≥ 0. Then there exist rational coefficients am and b` such that there holds X m Nv X `v−u (3.1) am(X − 1) = 1 + X b`X . K≤m≤(K+N)v−u 1≤`≤K The coefficients (b`) are unique and given explicitly by (−1)` Y u (3.2) b = j + N − . ` (` − 1)!(K − `)! v 1≤j≤K j6=` TITCHMARSH’S PROBLEM FOR MULTIPLICATIVE FUNCTIONS 11 Proof. An identity of the form (3.1) exists if and only if we can find b1, . . . , bK such that the first K − 1 derivatives of the polynomial on the right hand side of (3.1) vanish at X = 1. This is equivalent to saying that the b1, . . . , bK solve the equation 1 ··· 1 b1 −1 v + Nv − u ··· Kv + Nv − u b2 0 (3.3) = . . .. . . . . . . . . K−1 K−1 (v + Nv − u) ··· (Kv + Nv − u) bK 0 Let C be the matrix on the left, and B` the same matrix but with the upper row and the `-th column removed. Note that C is a Vandermonde matrix, and B` is a product of a Vandermond matrix with a diagonal matrix. Hence, we deduce Y K(K−1) det C = (jv − iv) = 2! 3! ··· (K − 1)! v 2 , 1≤i Since det C 6= 0, we obtain by Cramer’s rule that there is a unique solution (b`), given by det B (3.4) b = (−1)` ` , ` det C which yields (3.2). Theorem 3.2. Let v > 0 and r be integers such that v > u ≥ 0 and r ≥ 0. Let K ≥ 1 and x ≥ 1. Then for any n ≤ x, we have K X + X X (3.5) τr+ u (n) = c ··· τ 1 (n1) ··· τ 1 (n`v−u), v ` − v − v `=1 m1···m`+r n1···n`v−u=n 1/K n1,...,n`v−u≤x and, for r ≥ 1, K X − X X (3.6) τ−r+ u (n) = c ··· τ 1 (n1) ··· τ 1 (n`v+(r−1)v−u), v ` − v − v `=1 m1···m`−1n1···n`v+(r−1)v−u=n 1/K n1,...,n`v+(r−1)v−u≤x + − where the c` and c` are certain rational numbers, which can be bounded by + − c` , c` 1 for 1 ≤ ` ≤ K, the implicit constant depending only on K and r. Proof. Let ∞ ( 1/K X τ−1/v(n)g(n) 1 if n ≤ x , G(s) := with g(n) := ns 0 otherwise. n=1 1 We first look at (3.5). Here we use Lemma 3.1 with N = 0 and X = ζ(s) v G(s), and then multiply both r+ u sides by ζ(s) v , which leads to the identity X 1 m r+ u r+ u X r+` `v−u am(ζ(s) v G(s) − 1) ζ(s) v = ζ(s) v + b`ζ(s) G(s) . K≤m≤Kv−u 1≤`≤K Then (3.5) follows by comparing the Dirichlet coefficients on both sides. In order to show (3.6), we use Lemma 3.1 with the same X as before and with N = r − 1, and then −r+ u multiply both sides by ζ(s) v . This gives m X 1 −r+ u −r+ u X `−1 `v+(r−1)v−u am ζ(s) v G(s) − 1 ζ(s) v = ζ(s) v + b`ζ(s) G(s) , K≤m≤(K+r−1)v−u 1≤`≤K and (3.6) follows again by comparing the Dirichlet coefficients on both sides. Remark. With r = v = 1 and u = 0, the identity (3.6) leads to the decomposition of µ(n) described in [IK04, (13.38)]. 12 SARY DRAPPEAU AND BERKE TOPACOGULLARI 3.2. A combinatorial identity of Linnik’s type. Here we derive a combinatorial decomposition for τz using an approach analogous to [Lin63]. We denote by P +(n) the largest, and by P −(n) the smallest prime factor of an integer n > 1, with the convention that P +(1) = 1 and P −(1) = ∞. Given an arbitrary multiplicative function f and a complex number z ∈ C, we define the z-fold convolution of f as the multiplicative function given by X z X f (∗z)(pν ) := f(pλ1 ) ··· f(pλr )(ν ≥ 1). r 1≤r≤ν λ1,...,λr ≥1 λ1+···+λr =ν Note that f (∗z)(p) = zf(p), and that for ` ∈ N the `-fold convolution as defined here coincides with the `-fold convolution defined in the traditional sense. We will be eventually interested in the case (∗z) χ when f = χ is a Dirichlet character, in which case we have f = τz . Theorem 3.3. Let K ∈ N>0 and A, x ≥ 1. Then for all z ∈ C there exist complex numbers (c`)0≤`≤K , such that for all x ≥ 1 and all multiplicative functions f, we have the following identity for n ≤ x, (∗z) X X (∗(z−`)) (∗`) (3.7) f (n) = c` f (n1)f (n2), 0≤` z z X ` F (s) = F (s, y) c`G(s, y) + R(s), 0≤` Lemma 3.4. For any multiplicative function f : N → R, any compactly supported function g : N → C, and all y, w ≥ 2, we have X (3.8) f(n)g(n) = Σtriv + ΣI + O (ΣII) , P +(n)≤y where X X ΣI = f(n)g(n), Σtriv = f(n)g(n), n≤w n>w + P +(n)≤y P (n)≤y ∃pν kn,pν >y X X ΣII = (log y) sup αmβng(mn) , α,β w |αm| ≤ |f(m)|, |βn| ≤ (|f| ∗ |f|)(n). Proof. In an integer n with P +(n) ≤ y is not counted in the first two sums on the right-hand-side, then n > w and all prime powers pν kn satisfy pν ≤ y. By incorporating these prime powers as p increases, we may factor n = n1n2 uniquely in such a way that + − + P (n1) < P (n2), w < n1 ≤ wQ (n1), + + + ν where Q (n1) is the prime power corresponding to the largest prime of n1: Q (n1) = P (n1) kn1. Note that this implies (n1, n2) = 1. Our statement follows after separating variables [IK04, Lemma 13.11] in + − the condition P (n1) < P (n2). 4. Auxiliary estimates In this section we collect some estimates on ∆h(n, R), which will be needed in the following sections. 4.1. The second moment of ∆h(n; R). On several occasions, we will require the following rough upper-bound for the “main terms”. Lemma 4.1. For x ≥ 3, R ≥ 1 and (a, h) ∈ Z2 such that 1 ≤ a, |h|,R ≤ x1/4, the following estimate holds, X 2 2 4 |∆h(an; R)| τ((a, h)) x(log x) . x 2 X 2 X 2 X 2 |∆h(an; R)| τ(an − h) + |τeh(an; R)| =: G1 + G2, x x x 2 We start by splitting the sum according to the size of t∗ = (an − h, t∞) as follows, X X 2 X X 2 G1 = τ(an − h) + τ(an − h) =: G1a + G1b. ∗ ∞ x ∗ ∞ x t |t 2 t∗|t∞ a0x−2h0 a0x−h0 ∗ In G1b we bound all the summands trivially and get X X X 1 3 0 0 2 1+ε 4 +ε G1b τ(t(a n − h )) x ∗ x , ∗ ∞ x ∗ ∞ t t |t 2 Next we look at G2. Here we first rewrite τeh(an; R) as X X X 1 X h an τh(an; R) := 2 χ( αδ )χ( αδ ), e √ ϕ(q) α|(a,h) δ|( h ,n) an−h χ (mod q) α q≤ a αδ cond(χ)≤R (δ, α )=1 so that after expanding the square we are led to X X 1 X x G2 ≤ 4 S χ1χ2, [δ ,δ ] , √ ϕ(q1)ϕ(q2) 1 2 α1,α2|(a,h) ax−h χ1 (mod q1) h h q1≤ α δ δ1| , δ2| 1 1 χ2 (mod q2) α1 α2 √ ax−h cond(χ1),cond(χ2)≤R q2≤ α2δ2 with X S(χ, y) := max χ(n) . y 2 ≤y0 If χ1 and χ2 are induced by the same primitive character, we use the trivial bound S (χ1χ2, y) ≤ y. Otherwise, the Pólya-Vinogradov bound applies and S (χ1χ2, y) τ(q1q2)R log R. Inserting these bounds, we eventually obtain 2 4 ε 3 2 4 G2 τ((a, h)) x log x + x R τ((a, h)) x log x 1 by our assumption R ≤ x 4 . This concludes the proof. 4.2. Comparison of main terms. We begin by two technical lemmas related to the main terms that will appear later. Let X ≥ 1, and let f and v be two smooth functions which are both compactly ∗ supported inside R+. We assume that supp f ⊂ [C1X,C2X], where C1 and C2 are some fixed constants, and that for some Ω ∈ (0, 1], we have kv(j)k 1, kf (j)k (ΩX)−j |f (j+1)| (ΩX)−j, ∞ j ∞ j ˆ R TITCHMARSH’S PROBLEM FOR MULTIPLICATIVE FUNCTIONS 15 for all j ≥ 0. Furthermore, we define 1 X cd(h) ξ (4.2) M (b, h; M) := (log(ξ − h) + 2γ − 2 log d)f(ξ)v dξ, f,v b d ˆ bM d|b where X X cd(h) := e(νh/d) = δµ(d/δ) ν (mod∗ d) δ|(h,d) denotes the Ramanujan sum. Lemma 4.2. For (b, h) ∈ Z2, b, M ≥ 1, and R ≥ 1, we have X m (h, b) f(bm)v τ (bm; R) = M (b, h; M) + O XεR3/2 + X1/2+ε . M eh f,v b m Proof. Recall the definition (2.1). By partial summation and the Pólya-Vinogradov inequality, we have X m X m X 1 ε 3/2 f(bm)v τh(bm; R) = 2 f(bm)v q + O(X R ). M e M √ ϕ( ) m m q≤ bm−h (q,h) (bm,q)=(h,q) The condition (bm, q) = (h, q) in the sum on the right-hand side is equivalent to (h, q) m(b, q) q (b, q)|h, m, , = 1. (b, q) (h, q) (h, q) Using Möbius inversion and our hypotheses on f and v, we can replace the m-sum by the corresponding integral and obtain X m 2 ξ X (b, q) ε 3/2 f(bm)v τh(bm; R) = f(ξ)v dξ + O(X R ). M e b ˆ bM √ q m q≤ ξ−h (b,q)|h The main term on the right-hand side may be rewritten as √ 2 X cd(h) ξ ξ − h f(ξ)v H dξ + O(XεR3/2) b d ˆ bM d d|b P −1 where H(x) = q≤x 1/q = log x + γ + O(x ). This gives the claimed estimate. Next, we define χ X (4.3) Mf,v(b, h; M) := χ(a)Mfab,va/M (bD, h − ab; M/D) a mod D (a,D)=1 where fab(ξ) := f(ξ + ab) and va/M (ξ) := v(ξ + a/M). Lemma 4.3. If b = b◦b∗ with b◦|D∞ and (b∗,D) = 1, then 1 h b X cd(h) ξ − h ξ (4.4) M χ (b, h; M) = χ χ log + 2γ f(ξ)v dξ. f,v bD (h, b) (h, b) d ˆ (Db◦d)2 bM d|b∗ Moreover, if χ mod D is primitive, we have X m f(bm)v χ(m)τ (bm; R) M eh m X(log X)3 (h, b∗) = M χ (b, h; M) + O 1 (b, h) + XεD1/2R3/2 + X1/2+ε f,v D>R bD b∗ where 1D>R = 1 if D > R and 0 otherwise. Proof. We rewrite 1 X X cd(h − ab) ξ M χ (b, h; M) = χ(a) (log(ξ − h) + 2γ − 2 log d)f(ξ)v dξ. f,v bD d ˆ bM a (mod D) d|bD (a,D)=1 16 SARY DRAPPEAU AND BERKE TOPACOGULLARI Using Gauß sums, X −bD X hν χ(a)c (h − ab) = G(χ)χ χ(ν)e . d d d a (mod D) ν (mod∗ d) (a,D)=1 This last expression vanishes unless D(b, D∞)|d. Denoting b◦ = (b, D∞) and b∗ = b/b◦, we obtain for d|b∗ X ∗ X ◦ ◦ χ(a)cDb◦d(h − ab) = G(χ)χ(−b /d)G(χ) δχ(h/δ)µ(b d/δ)χ(b d/δ) a (mod D) δ|(b◦d,h) (a,D)=1 ◦ ∗ ◦ = b D1b◦|hχ(b )χ(h/b )cd(h) h b = b◦Dχ χ c (h). (h, b) (h, b) d This yields our first claim. For the second, the computations are similar to the previous Lemma. If D > R, we get X m (4.5) f(bm)v χ(m)τ (bm; R) XεD1/2R3/2, M eh m χ −1 2 while on the other hand Mf,v(b, h; M) (b, h)(bD) X(log X) by a simple computation from (4.4). If D ≤ R, the bound (4.5) applies to all the characters involved in the definition of τeh(bm; R), except all those which are induced by χ. We obtain h b X m X χ( (b,q) )χ( (b,q) ) X m f(bm)v χ(m)τ (bm; R) = 2 f(bm)v + O(XεD1/2R3/2). M eh ϕ( q ) M m q (h,q) D| (q,h) (bm,q)=(h,q) q2≤bm−h Similarly as above, the main term in the right-hand side can be rewritten 2 h b ξ X (q, b) χ χ f(ξ)v dξ + O(Xε) b (h, b) (h, b) ˆ bM √ q q≤ ξ−h q (b,q)|h, D| (h,q) (D,(b,h)/(b,q))=1 The χ-factors impose the conditions b◦|h and (D, h/b◦) = 1. We rewrite the q-sum as ∗ √ X (b, q) 1 X (q, b ) 1 X cd(h) ξ − h = = H q D q D d Db◦d √ √ ∗ q≤ ξ−h q≤ ξ−h/(Db◦) d|b q ∗ (b,q)|h, D| (h,q) (b ,q)|h (D,(b,h)/(b,q))=1 whence the claimed expression. 4.3. Type τ1 estimates. The following estimate is relevant for convolutions with one smooth compo- nent of size x1/3+ε. It can be viewed as a generalization of a result of Selberg [Sel91, p.235] on the equidistribution of τ2 in arithmetic progressions. Lemma 4.4. Let ε > 0, let C2 > C1 > 0, let v : (0, ∞) → R be a smooth and compactly supported function, and let χ mod D be a Dirichlet character of modulus D ≥ 1. Then we have, for any X,M ≥ 1 1−ε and R ≥ D, any 1 ≤ bD, |h| X , and any interval I ⊂ [C1X,C2X], X m ε 1 ∞ − 1 1 3 (4.6) χ(m)v ∆ (bm; R) X DX 3 + (b, hD )MX 2 + D 2 R 2 . M h m: bm∈I The implied constants depend only on the function v and the constants ε, C1 and C2. Proof. Note that we can always assume bM X, since otherwise the sums in consideration are empty. Let f : (0, ∞) → [0, ∞) be a smooth weight function, which is compactly supported in supp f ⊂ [C1X/2, 2C2X], which has value f(ξ) = 1 for all ξ ∈ I, and whose derivatives satisfy 1 1 f (ν)(ξ) for ν ≥ 0 and f (ν)(ξ) dξ for ν ≥ 1, (ΩX)ν ˆ (ΩX)ν−1 TITCHMARSH’S PROBLEM FOR MULTIPLICATIVE FUNCTIONS 17 for some constant Ω ≤ 1. We can then encode the condition bm ∈ I by using the function f(ξ) via X m X m (4.7) χ(m)v ∆ (bm; R) = f(bm)v χ(m)∆ (bm; R) + O ΩX1+εb−1 , M h M h m: bm∈I m so that it suffices to consider the smoothed sum on the right hand side. Assume first that χ is the trivial character. In [Top17b, Section 3] it is shown that X m ε 1 − 1 (4.8) f(bm)v τ(bm − h) = M (b, h; M) + O X b 2 Ω 2 , M f,v m where the main term Mf,v(b, h; M) is given by (4.2). By Lemma 4.2, we obtain X m ε 3 −1 1 +ε (4.9) f(bm)v τ (bm; R) = M (b, h; M) + O X R 2 + (b, h)b X 2 . M eh f,v m − 2 The estimate (4.6), in the case D = 1 and χ = 1, now follows from (4.7) with the choice Ω = bX 3 . Now assume that χ is a primitive character modulo D, where D ≤ R and bD X1−ε. We write ! X m X X m f(bm)v χ(m)τ(bm − h) = χ(a) fe(ebm)v τ(ebm − eh) , M e m a (mod D) m Mf (a,D)=1 with a eb := Db, Mf := M/D, eh := h − ab, fe(ξ) := f(ξ + ab) and v(ξ) := v ξ + , e M so that we can use our former result (4.8) to get X m χ ε 3 1 − 1 f(bm)v χ(m)τ(bm − h) = M (b, h; M) + O X D 2 b 2 Ω 2 , M f,v m χ where Mf,v(b, h; M) is defined in (4.3). By Lemma 4.3, we obtain X m X m f(bm)v χ(m)τ(bm − h) = f(bm)v χ(m)τ (bm; R) M M eh m m (h, b∗) + O XεD3/2b1/2Ω−1/2 + XεD1/2R3/2 + X1/2+ε ε b∗ − 2 We choose Ω = bDX 3 , and hence get (4.6) also in this case. The case when χ is not necessarily primitive follows at once using Möbius inversion. 4.4. Type τ2 estimates. The following estimate is a uniform version of the τ2 − τ2 shifted convolution problem obtained recently by the second author. Lemma 4.5. Let ε > 0, let C2 > C1 > 0, let v1, v2 : (0, ∞) → R be smooth and compactly supported weight functions, and let χ1 and χ2 be Dirichlet characters mod D. Then for any X, b ≥ 1 and R ≥ D, 1 1/4 any M1 ≥ M2 ≥ 1 with X 2 ≤ M1M2, any h ∈ Z with 1 ≤ |h|,D ≤ X and any interval I ⊂ [C1X/2,C2X], we have X m1 m2 (4.10) v v χ (m )χ (m )∆ (bm m ; R) 1 M 2 M 1 1 2 2 h 1 2 m1,m2: 1 2 bm1m2∈I |h|M M 1/4 b◦D5/2(XM M )1/3+ε 1 + 1 2 + X−1/2+εR3/2b◦(h, b)M M 2. 1 2 XD 1 2 The implied constant depends only on the constants ε, C1 and C2, and the functions v1 and v2. Proof. Note that we can make the assumption b X , as otherwise the sum in consideration is M1M2 empty. Also, as in Lemma 4.4, we can exchange the original sum by its smoothed version, X m1 m2 f(bm1m2)v1 v2 χ1(m1)χ2(m2)∆h(bm1m2; R), M1 M2 m1,m2 with an error of the size of O ΩX1+εb−1. 18 SARY DRAPPEAU AND BERKE TOPACOGULLARI Let χ0 := χ1χ2. The results of [Top17b] cannot be quoted as a black box, however, the computations of [Top17a] on which they are based may be adapted with little change. We write X m1 m2 X f(bm1m2)v1 v2 χ1(m1)χ2(m2)τ(bm1m2 − h) = χ1(a)D(a), M1 M2 m1,m2 a (mod D) where D(a) is the defined as X r1n + f1 r2n + f2 X D(a) := w1 w2 τ(r1n + f1) χ0(n2)hM2M1 (n1, n2), x1 x2 n n1,n2 n1n2=r2n+f2 with X r := bD, r := D, f := ab − h, f := a, x := X, x := , 1 2 1 2 1 2 b and p p n1 n2 w1(ξ) := f(Xξ + h), w2(ξ) := f(Xξ), hM2M1 (n1, n2) := v2 v1 . M2 M1 The sum D(a) is now of the same shape as the sum DAB(x1, x2) defined in [Top17a, p. 157], with the function fe(a, b) there replaced by χ0(a)fe(a, b). The computations of Section 3 of [Top17a] can then be 0 ± adapted with the following changes. In Section 3.1 of [Top17a], the expressions ΣAB and ΣAB have an ∗ ± additional factor χ0(au2/u2) in the summands. In the sums in the definition of RAB, p.159 ibid, the summand has to be multiplied by an additionnal factor χ0(c), and the altered relation ± X u2 X R Σ± = χ χ (d) AB AB 0 u∗ 0 d u∗|u 2 d 2 2 ∗ ∗ ∗ (d,r s2u )=1 r2 |r2 1 2 ± ± holds. Consequently, the relationship between RAB(N; χ) and KAB(N; χ) becomes ± X ± RAB(N; χ) = τ(n)Sbv(χ; n)KAB(χχ0; n). N we have X −1/2 B (4.11) (β ∗ γ)(n)∆h(an; R) A,η τ((a, h))R X(log X) . n∈I ε Proof. Recall that ∆h(an; R) RX . In the left-hand side of (4.11), the contribution of those n such that (n, (ah)∞) > Xδ is therefore at most X RXε 1 RX1−δ+ε. n X (n,(ah)∞)>Xδ Next, we have X X X X (β ∗ γ)(n)∆h(an; R) = βλ1mγλ2n∆h(aλ1λ2mn; R). ∞ ∞ −1 d|(ah) n∈I λ1,λ2|(ah) mn∈(λ1λ2) I δ d|n δ d≤X λ1λ2≤X (mn,ah)=1 (n/d,ah)=1 (m,λ2)=1 1/2+ε Finally, we note that there are at most O(X ) tuples (λ1, λ2, m, n) with λ1λ2mn ∈ I for which the expression aλ1λ2mn − h is a perfect square, and X X aλ1λ2 h ∆h(aλ1λ2mn; R) = 2 uR(mn λ λ ; q) + O(1aλ1λ2mn−h is a square), √ 3 3 λ3|(h,aλ1λ2) q≤ aλ1λ2mn−h/λ3 2 (q,aλ1λ2h/λ3)=1 where the notation uR(n; q) is defined in formula (5.1) of [Dra17]. Now, for each (λ1, λ2, λ3), the sum X X aλ1λ2 h S(λ1, λ2, λ3) = uR(mn ; q) λ3 λ3 √ −1 q≤ aλ1λ2mn−h/λ3 mn∈(λ1λ2) I 2 (q,aλ1λ2h/λ3)=1 (mn,ah)=1 (m,λ2)=1 is of the same shape as in formula (5.6) of [Dra17], with three differences: (1) the quantity τA(λ1)τA(λ2) has to be factored out for the condition (5.4) of [Dra17] to hold, (2) the sums over m and n must be restricted to dyadic intervals, which is done at the cost of an additionnal factor (log x)2, (3) the sums over m, n and q are not separated. The last point can be implemented by a standard argument (see e.g. page 720 of [Dra17]), cutting the (m, n) sums into intervals of type [M, (1 + ξ)M] × [N, (1 + ξ)N] with ξ R−1/2. Assuming δ is small enough in terms of η, we obtain −1 B −1 −1 S(λ1, λ2, λ3) τA(λ1)τA(λ2)(λ1λ2) X(log X) (ξ + ξ R ) −1 −1/2 B τA(λ1)τA(λ2)(λ1λ2) R X(log X) . We sum this over (λ1, λ2, λ3) satisfying ∞ δ λ1λ2|(ah) , λ1λ2 ≤ x , λ3|(h, aλ1λ2). P −1 OA(1) Since λ|(ah)∞ τ2A(λ)τ(λ)λ A (log log x) , we obtain X 1−δ/2 −1/2 B+1 (β ∗ γ)(n)∆h(an; R) τ((a, h)) RX + R X(log X) , n∈I which yields our claim by reinterpreting δ and B. 5. The case of rational parameters τ Q Let χ1, . . . , χT be distinct Dirichlet characters mod D, and the function f ∈ FD (A) be defined by ∞ T X f(n) Y (5.1) := L(s, χ )bj , ns j n=1 j=1 with b1, . . . , bT ∈ Q, which we write in the form uj bj = rj + with rj ∈ Z and uj, vj ∈ N such that 0 ≤ uj < vj. vj 20 SARY DRAPPEAU AND BERKE TOPACOGULLARI For notational convenience we also define X X krk1 := |rj|, kvk1 := vj. 1≤j≤T 1≤j≤T Our goal is to prove estimate (2.4) for the function f defined in (5.1). In fact, we will prove a result which is slightly more precise in term of uniformity in D and T . τ Q Proposition 5.1. Let A, D, T ≥ 1 be fixed. Then we have, for x ≥ 3, I ⊂ [x/2, x] and f ∈ FD (A) as described above, the following estimate, B+ω(D) 5 x(log x) 2 δ (5.2) |Σf (I; a, h; R)| ≤ Cτ((a, h))D 1 kvk1 for 1 ≤ a, |h|,R ≤ x , R 2 where δ > 0 is some absolute constant, and where B, C > 0 are constants which depend only on A and T . The rest of this section is now concerned with proving Proposition 5.1. χ 5.1. Application of the combinatorial identity. Denote τz (n) := τz(n)χ(n), so that (5.3) f(n) = τ χ1 ∗ · · · ∗ τ χT . b1 bT The expression on the left hand side of (5.2) now reads X (5.4) Σ (I; a, h; R) = τ χ1 (m ) ··· τ χT (m )∆ (am ··· m ; R). f b1 1 bT T h 1 T m1···mT ∈I By Theorem 3.2 with K = 4 we can write τ χj (m ) as bj j 4 χj X X X χj χj (5.5) τ (mj) = c`,j ··· χj(m1) ··· χj(mk )τ 1 (n1) ··· τ 1 (n 0 ), bj `,j − − k`,j vj vj `=1 m1···mk`,j n1···nk0 =mj `,j 1/4 n1,...,nk0 ≤x `,j 4 0 4 where (k`,j)`=1 and (k`,j)`=1 are two sequences of integers satisfying 0 0 ≤ k`,j ≤ |rj| + 4, 1 ≤ k`,j ≤ (|rj| + 4)vj, 4 and where (c`,j)`=1 is a set of complex numbers whose moduli are bounded in terms of A. We replace each factor τ χj (m ) in (5.4) by its decomposition, and after expanding the resulting expression, we end bj j up with a linear combination (whose coefficients are bounded by OA(1)) of OT (1) sums of the form X 0 (5.6) Ξ := σ1(m1) ··· σk(mk)%1(n1) ··· %k0 (nk0 )∆(am1 ··· mkn1 ··· nk; R), m1···mkn1···nk0 ∈I 1/4 n1,...,nk0 ≤x χj where each function σi is some Dirichlet character mod D, where each function %i is equal to τ for −1/vj some j, and where k and k0 are integers bounded by 0 X 0 ≤ k ≤ 4T + krk1 and 1 ≤ k ≤ 4kvk1 + |rj|vj. 1≤j≤T We consider each sum Ξ separately. Out of technical reasons, it will be necessary to use a smooth dyadic decomposition for the vari- ables m1, . . . , mk. Let u : (0, ∞) → R be a smooth and compactly supported function, which satisfies X ξ supp u ⊂ [1/4, 2] and u = 1 for all ξ ∈ (0, ∞), 2` `∈Z and define X ξ ξ u0(ξ) := u and u`(ξ) := u for ` > 0, M` M` `≤0 where we have set 1 +η ` M` := x 4 2 , 1 k with 0 < η < 24 an arbitrary, but fixed constant. For a k-tuple ` = (`1, . . . , `k) ∈ N , we then define X 0 0 0 Ξ` := u`1 (m1)σ1(m1) ··· u`k (mk)σk(mk)%1(n1) ··· %k (nk )∆(am1 ··· mkn1 ··· nk; R), m1···mkn1···nk0 ∈I 1/4 n1,...,nk0 ≤x TITCHMARSH’S PROBLEM FOR MULTIPLICATIVE FUNCTIONS 21 so that the sum Ξ can be split as X Ξ = Ξ`. k `∈N Note that this last sum is in fact finite, since Ξ` becomes empty if the coordinates of ` are large enough, namely if `1, . . . , `k log x. We will now estimate the sums Ξ` in different ways, depending on the sizes of the supports of the variables mi. 1 3 +η 5.2. Case I. First assume that ` has at least one coordinate, say `1, satisfying M`1 ≥ x . Let m0 := m2 ··· mkn1 ··· nk0 . Denoting σ1 = χj for some j, we can use Lemma 4.4 with X = ax, b = am0 and M = M`1 to get 1 1 1 3 X M`1 ε 3 3 ∞ 2 2 u`1 (m1)σ1(m1)∆h(am0m1; R) ε,A x Da x + (am0, hD ) 1 + D R . x 2 m1: m0m1∈I This leads to ε 1 1−η ∞ ω(D) 1 2 −η 1 3 (5.7) Ξ` ε x Da 3 x + (a, hD )(log x) x 2 + x 3 D 2 R 2 , where we have made use of the fact that ω(D) 1+ε X ∞ X ∗ X ε x X (log x) x (m0, hD ) ≤ D (m0, h) ε h 1 ε . x ∗ ∞ x M`1 ∗ ∞ M`1 m0≤ D |D m0≤ ∗ D |D M` D M` 1 D∗≤x 1 D∗≤x 5.3. Case II. Next assume that ` has at least two non-zero coordinates, say `1 ≥ `2 ≥ 1. We can also 1 1 4 +η 3 +η assume that x M`1 ,M`2 x , since the case of larger M`1 and M`2 is already treated above. Let m0 := m3 ··· mkn1 ··· nk0 . We use Lemma 4.5 with X = ax and b = am0, which gives X u`1 (m1)σ1(m1)u`1 (m1)σ2(m2)∆h(am0m1m2; R) m1,m2: m0m1m2∈I 5 1 3 M`1 M`2 ∞ ε 2 3 2 ε,A (am0,D )x D (axM`1 M`2 ) + (h, am0)R 1 1 , a 2 x 2 so that altogether we are led to ∞ ω(D) ε 5 1 1− 4 η 3 1 (5.8) Ξ` ε,A (a, h)(a, D )(log x) x D 2 a 3 x 3 + (a, h)R 2 x 2 . 5.4. Case III. Finally, we need to consider the case, where ` has at most one non-zero coordinate, 1 3 +η say `1, for which we have M`1 x . We split the sum Ξ` into two parts, (1) (2) Ξj =: Ξ` + Ξ` , η η according to whether n1 ··· nk0 > x or n1 ··· nk0 ≤ x . (1) We look first at Ξ` . We split this sum according to the value of 0 0 η µ = min{1 ≤ µ ≤ k : n1 ··· nµ0 > x }, and write accordingly k0 (1) X (1) Ξ` =: Ξ` (µ). µ=1 After defining X 0 0 βm := u`1 (m1)σ1(m1) ··· u0(mk)σk(mk)%µ+1(nµ+1) ··· %k (nk ), m1···mknµ+1···nk0 =m 1/4 nµ+1,...,nk0 ≤x and X γn := %1(n1) ··· %µ(nµ), n1···nµ=n η 1/4 n1···nµ−1≤x , n1,...,nµ≤x 22 SARY DRAPPEAU AND BERKE TOPACOGULLARI (1) and renaming n ← n1 ··· nµ and m ← m1 ··· mknµ+1 ··· nk0 , we can write Ξ` (µ) as (1) X Ξ` (µ) = βmγn∆h(amn; R). m,n: mn∈I xη |βm| ≤ τ2krk1+8T (m), |γn| ≤ τkrk1+4T (n). Hence we can apply Lemma 4.6 with A ← 2krk1 + 8T , and we see that (1) −1/2 B1 δ1 Ξ` (µ) τ((a, h))R x(log x) for 1 ≤ a, |h|,R ≤ x , where δ1,B1 > 0 are certain constants which depend solely on η and A. Summing over µ, we deduce (1) −1/2 B1 δ1 (5.9) Ξ` A τ((a, h))R x(log x) kvk1 for 1 ≤ a, |h|,R ≤ x . (2) The other sum Ξ` can be estimated similarly – the role of the variables n1, . . . , nk0 is now played by the variables m2, . . . , mk. Eventually, we get (2) −1/2 B2 δ2 (5.10) Ξ` A τ((a, h))R x(log x) for 1 ≤ a, |h|,R ≤ x , where δ2,B2 > 0 are certain constants which again depend solely on η and A. 5.5. Conclusion. Grouping the different bounds (5.7)–(5.10), setting B := max(B1,B2) and choos- ing δ > 0 small enough, we get 5 − 1 B+ω(D) δ Ξ τ((a, h))D 2 R 2 x(log x) kvk1 for 1 ≤ a, |h|,R ≤ x , with the implicit constant depending only on A and T . This finally proves Proposition 5.1. 6. Interpolation to complex parameters ω Let r1, . . . , rϕ(D) be the residues mod D which are relatively prime to D. Any f ∈ FD(A) is given by ∞ ϕ(D) X f(n) Y Y zj (6.1) = 1 + , ns ps − 1 n=1 j=1 p≡rj mod D ϕ(D) for z = (z1, . . . , zϕ(D)) ∈ C , with |zj| ≤ A. After setting (6.2) ωr(n) := # {p prime : p | n, p ≡ r mod D} , we can also write ϕ(D) X Y ωr (nj ) f(n) = zj j . n1···nϕ(D)=n j=1 ω Our aim here is to show that the bound (2.4) holds for Σf (I; a, h; R), for all f ∈ FD(A). By Lemma 2.5 this will imply Proposition 2.1. Let χ1, . . . , χϕ(D) be the Dirichlet characters mod D, let Q be the unitary matrix χ1(r1) χ2(r1) ··· χϕ(D)(r1) 1 χ1(r2) χ2(r2) ··· χϕ(D)(r2) Q := , p . . .. . ϕ(D) . . . . χ1(rϕ(D)) χ2(rϕ(D)) ··· χϕ(D)(rϕ(D)) ϕ(D) ϕ(D) and let MQ : C → C be the bijective linear map associated to Q. ω ω Let K ≥ 1. We define FD(A, K) to be the set of functions f ∈ FD(A) of the same form as in (6.1), but with the additional property that the parameters z are given by z = MQ(b) ϕ(D) for a tuple of rational numbers b = (b1, . . . , bϕ(D)) ∈ Q satisfying uj |bj| ≤ A and bj = with uj, vj ∈ Z and |vj| ≤ K, vj for all j = 1, . . . , ϕ(D). By Proposition 5.1, Lemma 2.2 and Lemma 2.3, we deduce that the bound (2.4) ω holds for all f ∈ FD(A, K) in the following form. TITCHMARSH’S PROBLEM FOR MULTIPLICATIVE FUNCTIONS 23 ω Proposition 6.1. Let A, D ≥ 1 be fixed. For K ≥ 1, x ≥ 3, I ⊂ [x/2, x] and f ∈ FD(A, K), we have B x(log x) δ (6.3) |Σf (I; a, h; R)| ≤ CKτ((a, h)) 1 for 1 ≤ a, |h|,R ≤ x , R 2 where δ > 0 is some absolute constant, and where B, C > 0 are constants which depend only on A and D. ω ω Our goal is to interpolate this result to all functions in FD(A). Let f ∈ FD(A) be fixed, with z as in (6.1). For L ∈ [1, ∞], we define two polynomials in the variables Z = (Z1,...,Zϕ(D)) as follows, ϕ(D) X X Y ωr (nj ) PL(Z) := Zj j ∆h(an; R), PeL(Z) := PL(MQ(Z)). n∈I n1···nϕ(D)=n j=1 ∀j,ωrj (n)≤L By definition, both these polynomials have degree at most L in each variable. Furthermore, let −1 b := MQ (Z), 1 and note that kbk∞ ≤ D 2 A. Using this notation, we can now write the sum Σf (I; a, h; R) simply as X f(n)∆h(an; R) = Pe∞(b). n∈I In order to have better control over the degree of Pe∞(Z), we cut off all the terms of degree larger 1 than some fixed real number L ≥ 1. For a tuple ζ = (ζ1, . . . , ζϕ(D)) satisfying |ζj| ≤ AD 2 and any real number E ≥ 1, this leads to an error term of the following form, X ω(n) |Pe∞(ζ) − PeL(ζ)| ≤ τD(n)(AD) |∆h(an; R)| n∈I ω(n)>L −L X ω(n) ≤ E τD(n)(ADE) |∆h(an; R)| n≤x 1 1 2 2 −L X 2 X 2 ≤ E τAD2E(n) |∆h(an; R)| . n≤x n≤x The different factors can be estimated via [Ten15, Theorem II.6.1], and Lemma 4.1, and we get 1 4 2 1 −L (ADE) −1 4 2 2 |Pe∞(ζ) − PeL(ζ)| E x(log x) x(log x) τ((a, h)) 4 −L (ADE) +2 (6.4) E τ((a, h))x(log x) 2 , where the implicit constants depend at most on A, E and D. Next, we set 1 2(` + 1)bAD 2 c 1 β := − bAD 2 c for ` = 0, . . . , L. ` L + 1 1 Obviously, all these numbers are bounded by |β`| ≤ AD 2 , and are rational numbers with denominators not larger than L + 1. Furthermore, we have the bound 1 AD 2 |β − β | ≥ |` − ` | for ` 6= ` . `1 `2 2L 1 2 1 2 ϕ(D) For any tuple ` = (`1, . . . , `ϕ(D)) ∈ {0,...,L} , denote β` = (β`1 , . . . , β`ϕ(D) ). The value Pe∞(β`) can be interpreted as an instance of the sum Σ (I; a, h; R) for an appropriate function fe ∈ F ω (AD, L + 1), fe D Pe∞(β`) = Σ (I; a, h; R). fe Hence, by Proposition 6.1 and the estimate in (6.4) we can deduce (ADE)4 L 1 2 +B (6.5) PeL(β`) A,D τ((a, h))x(log x) 1 + L , R 2 E uniformly for 1 ≤ a, |h|,R ≤ xδ. 24 SARY DRAPPEAU AND BERKE TOPACOGULLARI By Lagrange interpolation, we bring PeL(b) into the following shape, ϕ(D) X Y Y bj − βi PeL(b) = PeL(β`) , β` − βi `∈{0,...,L}ϕ(D) j=1 0≤i≤L j i6=`j which is allowed since the Vandermonde determinant associated to (β`) does not vanish. We can now estimate PeL(b) via the already known bound (6.5) for the expressions PeL(β`). Namely, we have ϕ(D) X Y Y |bj − βi| |PfL(b)| ≤ PeL(β`) |β` − β` | `∈{0,...,L}ϕ(D) j=1 0≤i≤L j i i6=`j ϕ(D) 4 (ADE) L 1 X Y Y 1 2 +B Lϕ(D) τ((a, h))x(log x) 1 + L (4L) R 2 E |`j − i| `∈{0,...,L}ϕ(D) j=1 0≤i≤L i6=`j 4 Lϕ(D) (ADE) +B L 1 (8L) τ((a, h))x(log x) 2 + , 1 L ϕ(D) R 2 E (L!) which after using Stirling’s approximation for the Gamma function simplifies to (ADE)4 1 1 2 +B 2DL |PfL(b)| τ((a, h))x(log x) 1 + L (4e) , R 2 E with the implicit constant depending at most on A, E and D. After adding all the terms we had cut off earlier, we are finally led to (ADE)4 1 1 2 +B 2DL Σf (I; a, h; R) A,D,E τ((a, h))x(log x) 1 + L (4e) . R 2 E With the choices log R L := and E := (4e)6D, 12D log(4e) and after reinterpreting the constant B, we get B x(log x) δ Σf (I; a, h; R) A,D τ((a, h)) 1 for 1 ≤ a, |h|,R ≤ x , R 3 which is exactly the statement we wanted to prove. 7. Proof of Theorem 1.2 using Linnik’s identity We now sketch how Theorem 1.2 can alternatively be proven using Theorem 3.3. The details of the computations being very similar, we will restrict to discussing the main differences in the arguments. τ As mentioned above, it is enough to consider the case f ∈ FD(A), or in other words we can assume that f = τ χ1 ∗ · · · ∗ τ χT , where χ , . . . , χ are distinct Dirichlet characters mod D, and where b , . . . , b b1 bT 1 T 1 T are complex numbers whose moduli are bounded by A. The sum in consideration is then given by X Σ (I; a, h; R) = τ χ1 (m ) ··· τ χT (m )∆ (am ··· m ; R). f b1 1 bT T h 1 T m1···mT ∈I Here we replace each τ χj (m ) by its decomposition as given in Theorem 3.3 with K = 4, and after bj j expanding the resulting expression, we end up with a linear combination of sums of the form X Ξ := σ1(m1) ··· σk(mk)ρ1(n1) ··· ρT (nT )∆(am1 ··· mkn1 ··· nT ; R), m1···mkn1···nT ∈I + 1/4 P (n1···nT )≤x where each function σ is some Dirichlet character mod D, where each function ρ is equal to τ χj for j j bj −` some j and ` ∈ [0, 3], and where k ≤ 3T . We consider each sum Ξ separately. 1/4 η To each factor ρj in the sum Ξ we apply Lemma 3.4 with y = x and w = x for some arbitrary, but fixed η ∈ (0, 1/24). By compacity, it follows that for each j = 1,...,T there exist arithmetic functions αj and βj, such that the sum Ξ can be written as T T X (1) X (2) (3) Ξ = Ξj + Ξj + Ξ , j=1 j=1 TITCHMARSH’S PROBLEM FOR MULTIPLICATIVE FUNCTIONS 25 with (1) X Ξj := (σ1 ∗ · · · ∗ σk)(m)ρ1(n1) ··· ρT (nT )∆(amn1 ··· nT ; R), mn1···nT ∈I + 1/4 P (n1···nT )≤x η 1/η n1,...,nj−1≤x , nj >x k k 1/4 ∃p knj , p >x ! (2) X Y 0 00 Ξj := (σ1 ∗ · · · ∗ σk)(m) ρk(nk) αj(nj)βj(nj )∆(amn1 ··· nT ; R), 0 00 1≤k≤T mn1···nj1 nj nj nj+1···nT ∈I + 0 00 1/4 k6=j P (n1···nj1 nj nj nj+1···nT )≤x η η 0 1/4+η n1,...,nj−1≤x , x mn1···nT ∈I η n1,...,nT ≤x (1) k The sums Ξj can be bound trivially. Indeed, we note that if a prime power p > y divides n, then + since P (nj) ≤ y we must have k ≥ 2. Hence (1) X Ξj ≤ τ(A+6)T (n)|∆(an; R)| n∈I ∃pk|n: pk>x1/4, k≥2 1 1 ! 2 ! 2 X 2 X ε |∆(an; R)| 1 n∈I n∈I ∃pk|n: pk>x1/4, k≥2 1−1/17 A,T x τ((a, h)), which is an acceptable error term. (2) Concerning the sums Ξj , we can bound them following the arguments of Case III, Section 5.4, since we have a variable localized in [xη, x1/4+η], and since 1/4 + η < 1/3. The remaining sum Ξ(3), which is analogous to (5.6), can be estimated for all sufficiently small η > 0 by the arguments of Sections 5.2, 5.3 and 5.4, according to the size of the involved variables. As a result, we get for these sums the estimate O(1) (2) (3) x(log x) Ξj , Ξ A,T τ((a, h)) 1 . R 2 (1) Together with the bound for Ξj , this eventually proves Theorem 1.2. 8. Proof of Theorems 1.2, 1.3, 1.4 and 1.6 In this section we want to deduce Theorem 1.2 from Proposition 2.1, and afterwards apply this result to the problems mentioned in the introduction. Before doing so, we first need to prove an auxiliary result, which is concerned with bounds on average for functions in FD(A) twisted by a Dirichlet character. Lemma 8.1. Let f ∈ FD(A) and let B ≥ 1. Then there exists a constant c > 0, such that, for all Dirichlet characters χ mod q satisfying cond(χ) - D and q ≤ (log x)B, we have √ X (8.1) χ(n)f(n) xe−c log x. n≤x Both the constant c and the implicit constant depend at most on A, B and D. Proof. Let Fχ(s) be the Dirichlet series associated to the function χ(n)f(n). By Lemma 2.4 we know that Fχ(s) can be written as Y bψ Fχ(s) = Hχ(s) L(s, χψ) for Re(s) > 1, ψ mod D 1 where Hχ(s) is a holomorphic function in Re(s) ≥ 2 + ε, bounded in terms of A, D only. Due to the assumption cond(χ) - D we know that none of the characters χψ is principal, which means that none of the L-functions L(s, χψ) has a pole at s = 1. It follows from Siegel’s theorem that 26 SARY DRAPPEAU AND BERKE TOPACOGULLARI for any δ > 0 there exists a constant c(δ) such that all L(s, χψ) are zero-free in the region defined by the condition Re(s) > 1 − γ(Im(s)), where c(δ) c(δ) (8.2) γ(t) := min , . log(qD(|t| + 2)) (qD)δ Using this zero-free region, the bound (8.1) follows using a standard contour integration argument; see e.g. [MV07, Section 11.3]. We now proceed to prove Theorem 1.2. We set R = (log x)L where L ≥ 1 is some constant which depends only on A, B and D, and which we will determine at the very end. Note that in any case we can assume x to be large enough so that D ≤ R is satisfied. We start by splitting the sum Df (x; a, h) into two parts as follows, √ X Df (x; a, h) = Df ( x; a, h) + f(n)τ(an − h). √ x (2) X X X 1 X h a tu2q2 Mf (x; a, h) = 2 χ tu χ t Sf,χ(x, u) − Sf,χ a , u , √ ϕ(q) t|(a,h) u| h ax χ mod q t q≤ tu (u,a/t)=1 cond χ≤R cond(χ)-D with Sf,χ(x, u) given by X Sf,χ(x, u) := f(un)χ(n). x n≤ u This last sum can be estimated via Lemma 8.1, namely we have X ∗ ∗ X 1 +ε Sf,χ(x, u) = f(uu )χ(u ) f(n)χ(n) + O x 2 u∗|u∞ n≤ x √ uu∗ u∗≤ x (n,u)=1 √ ∗ X τA(uu ) 1 −c log x 2 +ε xe ∗ + x √ uu u∗≤ x τ (u) √ A x(log x)Ae−c log x, u for some constant c > 0 depending on A, D and L. Hence √ √ (2) c A+2 −c log x − 2 log x Mf (x; a, h) τ((a, h))Rx(log x) e τ((a, h))xe . Eventually, we get B √ (log x) − c log x D (x; a, h) = M (x; a, h) + O τ((a, h))x + e 2 , f f R1/3 TITCHMARSH’S PROBLEM FOR MULTIPLICATIVE FUNCTIONS 27 and Theorem 1.2 follows with the choice L = 3N + 3B. 8.1. Proof of Theorems 1.3, 1.4 and 1.6. The applications mentioned in the introduction are essentially all immediate corollaries of Theorem 1.2, except for the fact that it remains to evaluate the main terms. This is a rather tedious task, but can be done using standard techniques from analytic number theory, in particular the Selberg-Delange method, which is for example described in detail in [Ten15, Chapter II.5]. In order to not further lengthen this article, we only want to indicate very briefly the main steps of the procedure. In the case of Theorem 1.3, the main term takes the form X 1 X Mτz (x; 1, h) = 2 τz(n), √ q q≤ x ϕ (h,q) q2≤n≤x (n,q)=(h,q) which after a few simple transformations can be written as 2 2 X X D(x; uq, uv) − D(u q ; uq, uv) 1 2 +ε (8.3) Mτz (x; 1, h) = τz(uv) + O x , √ ϕ(q) u|h, v|u∞ √ q≤ x/u v≤ x vh (q, u )=1 where X D(y; r, t) := τz(n). y n≤ t (n,r)=1 This sum has been studied in detail in [Ten15, Chapter II.5]. In particular, following the proof of [Ten15, z Theorem II.5.2], we see that there exist complex numbers µ` (r, t) such that L 1 X µz(r, t) y(log y)z (log t)L+1 y(log y)z D(y; r, t) = ` + O , 2πi Γ(z − `) (log y)`+1 t (log y)L+2−ε `=0 where z ψz(r) (s − 1)zζ(s)z Y 1 µz(r, t) := ∆` s with ψz(r) := 1 − , ` s ts s s ps p|r ` and where the differential operator ∆s is defined as ` ` 1 ∂ ∆s := ` . `! ∂s s=1 It therefore remains to evaluate the sums ` z ` z 2 z X ∆sψs (uq) X ∆sψs (uq) q (2 log(uq)) and `−1 . √ ϕ(q) √ ϕ(q) (2 log(uq)) x x q≤ u q≤ u vh (q,vh/u)=1 (q, u )=1 For the first sum this is a standard exercise in using counter integration, the result being ` z z vh w ! X ∆ ψ (uq) ψs (u)ρw x 2 ζ(w + 1) 1 s s = ∆` Res Cz u + O , s s,w z w 2 −ε √ ϕ(q) w=0 γ (h) u w 3 x s,w x q≤ u vh (q, u )=1 with Y 1 ψz(p) − 1 Cz := 1 + + s , s,w (p − 1)pw+1 (p − 1)pw p and Y p(ψz(p) − 1) Y p γz (n) := 1 + s and ρ (n) := 1 − . s,w pw+2 − pw+1 + 1 w pw+2 − pw+1 + 1 p|n p|n An asymptotic formula for the second sum now follows via partial summation. After putting the resulting formulae back in (8.3) and completing the sum over v, this eventually leads to the main term described 28 SARY DRAPPEAU AND BERKE TOPACOGULLARI in Theorem 1.3. In particular, the first coefficient is given by 1 z−1 ! 1 Y 1 − p − 1 λ (z) = 1 + h,0 Γ(z) p (p,h)=1 (8.4) z+1 `−1 j z−1 ` ! Y 1 1 X (` − j)τz(p ) 1 τz(p ) · 1 − + 1 − + 1 − . p p pj p p`+1 p`kh j=1 For Theorem 1.5, we have from [Nar04, Proposition 8.4, Theorem 8.6] that the characteristic func- P tion n 7→ bK (n) of the set NK is multiplicative with b(p) = 1 if and only if χ∈X(K) χ(p) > 0, where X(K) is a subgroup of the Dirichlet characters modulo the discriminant D = Disc(K) and p - D. P The subgroup of residue classes a mod D such that χ∈X(K) χ(a) > 0, corresponding to the subgroup H in [Nar04, Theorem 8.2], has density 1/[K : Q] inside (Z/DZ)×. Thus we have a factorization X bK (n) = ζ(s)1/[K:Q]H(s) ns n≥1 2 where H is holomorphic and bounded in the strip Re(s) ≥ 3 . The rest of the argument the follows the path described above. We leave the details to the reader. In the case K = Q(i), the first coefficient is given by βh,0 = B0B(h), where − 1 1 Y 1 2 (8.5) B0 := √ 1 − , 2 p2 p≡3 mod 4 and ∗ ` χ4(h ) Y 1 (−1) Y 1 B(h) := 1 + 1 − + 1 + , 4h◦ p + 1 p`(p + 1) p2 p`kh p≡3 mod 4 p≡3 mod 4 ◦ ∞ ∗ h with h := (h, 2 ), h := h◦ and χ4 the non-principal character mod 4. Finally, the proof of Theorem 1.6 rests upon the fact that X 1 ∂k X τ(n − h) = Ξ (0) with Ξ (z) := zω(n)τ(n − h). k! ∂zk x,h x,h |h| L Re(z) X γh,`(z) x(log x) Ξ (z) = x(log x)z + O . x,h (log x)` (log x)L+1−ε `=0 Now an application of [Ten15, Theorem II.6.3] proves Theorem 1.6. References [ABSR15] J. C. Andrade, L. Bary-Soroker, and Z. Rudnick. Shifted convolution and the Titchmarsh divisor problem over Fq[t]. Philos. Trans. Roy. Soc. A, 373(2040):20140308, 18, 2015. Erratum, ibid. 374(2060), 2015. [BFI86] E. Bombieri, J. B. Friedlander, and H. Iwaniec. Primes in arithmetic progressions to large moduli. Act. Math., 156:203–251, 1986. [Bom65] E. Bombieri. On the large sieve. 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Math. Soc. Transl. Ser. 2, 82:9–46, 1969. Aix Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373, 13453, Marseille, France E-mail address: [email protected] EPFL SB MATH TAN, Station 8, 1015 Lausanne, Switzerland E-mail address: [email protected]