Sums of Kloosterman Sums in Arithmetic Progressions, and the Error Term in the Dispersion Method Sary Drappeau

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Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method Sary Drappeau

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Sary Drappeau. Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method. Proceedings of the London Mathematical Society, London Mathematical Society, 2017, 114 (4), pp.684-732. 10.1112/plms.12022. hal-01302604

HAL Id: hal-01302604 https://hal.archives-ouvertes.fr/hal-01302604 Submitted on 14 Apr 2016

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. arXiv:1504.05549v3 [math.NT] 14 Dec 2015 fcretmtos ow consider we so methods, current of generally, oiae ytoisacso h usin(.) h Titchm the functions. divisor p (1.1): of question of the correlation function of characteristic instances two the by motivated or instance, for functions rtmtcporsin fmdl pto up moduli of progressions arithmetic fdvsrlk ucin,frisac et-rw’ ide Heath-Brown’s instance for functions, divisor-like of a olo tti usini otyt siaesm ftety the of sums estimate to try to is (1.1) question this at look to way when nees tdigtefunction the Studying integers. s( as for (1.2) where 1.1. hc sdee u oedffiutt banas obtain to difficult more but deeper is which nwa follows, what In Date 1 2010 nesadn h on utpiaiesrcueo pairs of structure multiplicative joint the Understanding hr r ubro omlsrltn h hrceitcf characteristic the relating formulas of number a are There k ,n n, h icmrhdvsrproblem. divisor Titchmarsh The ≥ ,g f, eebr1,2015. 15, December : p ahmtc ujc Classification. Subject Mathematics oko ehulesadIaic n skyt bann pow obtaining method. to dispersion key the is notably and applications, Iwaniec, Th in and variables. Deshouillers summation of “smooth” work the on conditions gruence h xoeti hw ob needn of independent be to different shown a is by exponent Vinogradov the Bykovski˘ı and by announced result Abstract. assumed. banpwrsvn nteaypoi oml for formula Fou asymptotic of the work in Extending power-saving negative. obtain always is it showing zeroes, epoepwrsvn ro emi h icmrhdvsrp divisor Titchmarsh the in term error ing power-saving prove we 1 sa usadn rbe nmlilctv ubrtheo number multiplicative in problem outstanding an is +1) ,tesum the 2, eoe primes. denotes sacneune suigteReanhptei o Diric for hypothesis Riemann the assuming consequence, a As USO LOTRA USI ARITHMETIC IN SUMS KLOOSTERMAN OF SUMS : τ P k N ( p n ≤ → eoe h ubro asoecnwrite can one ways of number the denotes ) x RGESOS N H RO TERM ERROR THE AND PROGRESSIONS, τ ( C p epoeabudfrqitlna uso lotra us w sums, Kloosterman of sums quintilinear for bound a prove We − r w ucin htaeo utpiaientr multip – nature multiplicative of are that functions two are τ ( ) nodtoal,w slt h osbecontribution possible the isolate we Unconditionally, 1). n NTEDSESO METHOD DISPERSION THE IN eoe h ubro iioso h integer the of divisors of number the denotes ) priori A τ k hswudrqieudrtnigpie pto up primes understanding require would this , 1. ie oeisgtit h atrsto fnumbers of factorisation the into insight some gives AYDRAPPEAU SARY n X k ≤ Introduction p X x 2. = ≤ 10 piay,1F0 17,11N13. 11N75, 11F30, (primary), 11L07 f x ( τ n x k 1 ) ( 1 − g p 1 ( − k /k n n ol iet eal oevaluate, to able be to like would One fagnrlzdLneö yohssis hypothesis Lindelöf generalized a if h case The . 1) + 1) k tt [HB82]. ntity grows. P nto fpie olna combination linear to primes of unction n ≤ x τ k ie.I hspprw are we paper this In rimes. ( fnihoigitgr such integers neighboring of k n r n eebu,we Tenenbaum, and vry n ) sgnrlzsclassical generalizes is ≥ rhdvsrpolm and problem, divisor arsh sapoutof product a as ehd h anin gain The method. τ rsvn ro terms error er-saving ( olmo estimat- of roblem n em a rmreach from far seems 3 pe ) ervn a reproving 1), + hlet y quantitative A ry. L -functions, fSiegel of t con- ith n n more and , k positive licative x in 1 , 2 SARY DRAPPEAU

In place of (1.2), one may consider T (x) := Λ(n)τ(n 1) 1 1 under the assumption that the Riemann hypothesis∼ holds for all Dirichlet L-functions. This asymptotics was proved unconditionally by Linnik [Lin63] using his so-called dispersion method. Simpler proofs were later given by Rodriquez [Rod65] and Hal- berstam [Hal67] using the theorems of Bombieri-Vinogradov and Brun-Titchmarsh. Finally the most precise known estimate was proved independently by Bombieri– Friedlander–Iwaniec [BFI86] and Fouvry [Fou85]. To state their result, let us denote 1 log p C1 := 1+ , C2 := . p p(p 1) p 1+ p(p 1) Y − X − Theorem A (Fouvry [Fou85], Bombieri–Friedlander–Iwaniec [BFI86]). For all A> 0 and all x 3, ≥ T (x)= C x log x +2γ 1 2C + O x/(log x)A . 1 − − 2 A n o In this statement, γ denotes Euler’s contant. See also [Fel12, Fio12a] for generaliza- tions in arithmetic progressions; and [ABSR15] for an analog in function fields. The error term in Theorem A is due to an application of the Siegel-Walfisz theorem [IK04, Corollary 5.29]. One could wonder whether assuming the Riemann Hy- pothesis generalized to Dirichlet L-functions (GRH) allows for power-saving error term to be obtained (as is the case for the prime number theorem in arithmetic progressions [MV07, Corollary 13.8]). The purpose of this paper is to prove that such is indeed the case. Theorem 1.1. Assume GRH. Then for some δ > 0 and all x 2, ≥ 1 δ T (x)= C x log x +2γ 1 2C + O(x − ). 1 − − 2 n o Unconditionally, we quantify the influence of hypothetical Siegel zeroes. Define, for q 1, ≥ 1 1 log p C (q) := 1+ , C (q) := 1 ϕ(q) p(p 1) 2 1+ p(p 1) Yp∤q − Xp∤q − where ϕ is Euler’s totient function. Note that C1 = C1(1) and C2 = C2(1). Theorem 1.2. There exist b> 0 and δ > 0 such that T (x)= C x log x +2γ 1 2C 1 − − 2 n β o x x 1 δ√log x C (q) log +2γ 2C (q) + O xe− . − 1 β q2 − β − 2 n o The second term is only to be taken into account if there is a primitive character χ (mod q) with q e√log x whose Dirichlet L-function has a real zero β with β 1 b/√log x. ≤ ≥ − By partial summation, one deduces ERROR TERM IN THE DISPERSION METHOD 3

Corollary 1.3. In the same notation as Theorem 1.2, β x β δ√log x τ(p 1) = C1 x+2 li(x)(γ C2) C1(q) +2 li(x )(γ log q C2(q)) +O(xe− ). p x − { − }− { β − − } X≤ The method readily allows for more general shifts τ(p a), 0 < a xδ (cf. [Fio12b, Corollary 3.4] for results on the uniformity in a). In the− case a =| 1,| or≤ more generally when a is a perfect square, we have an unconditional inequality. Corollary 1.4. With an effective implicit constant, we have δ√log x τ(p 1) C1 x + 2 li(x)(γ C2) + O(xe− ). p x − ≤ { − } X≤ We conclude our discussion of the Titchmarsh divisor problem by mentioning the 1 δ important work of Pitt [Pit13], who proves p x a(p 1) x − for the sequence (a(n)) ≤ − ≪ of Fourier coefficients of an integral weightP holomorphic cusp form (which is a special case of (1.1) when the (a(n)) are Hecke eigenvalues). It is a striking feature that power-saving can be proved unconditionally in this situation. 1.2. Correlation of divisor functions. Another instance of the problem (1.1) is the estimation, for integers k,ℓ 2, of the quantity ≥ k,ℓ(x) := τk(n)τℓ(n + 1). T n x X≤ The conjectured estimate is of the shape k+ℓ 2 (x) C x(log x) − Tk,ℓ ∼ k,ℓ for some constants Ck,ℓ > 0. The case k = ℓ is of particular interest when one looks at the 2k-th moment of the Riemann ζ function [Tit86, §7.21] (see also [CG01]): in that context, the size of the error term is a non-trivial issue, as well as the uniformity with which one can replace n + 1 above by n + a, a = 0. Current methods are ineffective when k,ℓ 3, so we focus on the case ℓ = 2. Let6 us denote ≥ k(x) := τk(n)τ(n + 1). T n x X≤ There has been several works on the estimation of k(x). There are nice exposi- tions of the history of the problem in the papers of Heath-BroT wn [HB86] and Fouvry- Tenenbaum [FT85]. The latest published results may be summarized as follows. Theorem B. There holds: (x)= xP (log x)+ O (x2/3+ε), ([DI82a]), T2 2 ε 1 δ (x)= xP (log x)+ O(x − ), ([Des82], [Top15]), T3 3 δ√log x (1.3) (x)= xP (log x)+ O (xe− ) for fixed k 4, ([FT85]). Tk k k ≥ Here ε> 0 is arbitrary, δ > 0 is some constant depending on k, and Pk is an explicit degree k polynomial. The error term of (1.3) resembles that in the distribution of primes in arithmetic progressions, where it is linked to the outstanding problem of zero-free regions of L- functions. However there is no such process at work in (1.3), leaving one to wonder if power-saving can be achieved. In [BV87], Bykovski˘ıand Vinogradov announce results implying 1 δ/k (1.4) (x)= xP (log x)+ O (x − ) (k 4, x 2) Tk k k ≥ ≥ 4 SARY DRAPPEAU for some absolute δ > 0, and sketch ideas of a proof. The proposed argument, in a way, is dual to the method adopted in [FT85]2 (which is related to earlier work of Motohashi [Mot76]). Here we take up the method of [FT85] and prove an error term of the same shape. Theorem 1.5. For some absolute δ > 0, the estimate (1.4) holds. In view of [BV87], Theorem 1.5 is not new. However the method is somewhat different. In the course of our arguments, the analytic obstacle to obtaining an error 1 δ term Ok(x − ) (δ independent of k) in the estimate (1.4) will appear clearly: it lies in the estimation of sums of the shape n x τk(n)χ(n) for Dirichlet characters χ of ≤ small conductors. This issue is know to beP closely related to the growth of Dirichlet L- functions inside the critical strip [FI05]. Theorem 1.6. Assume that Dirichlet L-functions satisfy the Lindelöf hypothesis, 1 ε meaning L( 2 + it, χ) ε (q( t + 1)) for t R and χ (mod q). Then for some absolute δ > 0, ≪ | | ∈ 1 δ (1.5) (x)= P (log x)+ O (x − ) (k 4, x 2) Tk k k ≥ ≥ 1/2+ε The standard conjecture for the error term in the previous formula is Ok,ε(x ). We have not sought optimal values for δ in Theorems 1.5 and 1.6. In the case of (1.4), the method of [BV87] seems to yield much better numerical results. Our method readily allows to replace the shift n + 1 in Theorem 1.5 by n + a, 0 < a xδ with an exponent independent of k. We give some explanations in Section| | 7.3 ≤ below regarding this point. Acknowledgements. This work was done while the author was a CRM-ISM Post- doctoral Fellow at Université de Montréal. The author is indebted to R. de la Bretèche, É. Fouvry, V. Blomer, D. Milićević, S. Bettin and B. Topacogullari for valuable dis- cussions on the topics in this paper, and to G. Tenenbaum for helpful comments on an earlier version of the manuscript. The author is particularly grateful to V. Blomer for making a preprint of [BM15a] available, and for making him aware of the reference [BV87]; and finally to B. Topacogullari for correcting a significant oversight in the statement of the trace formula and large sieve inequalities in a previous version. 2. Overview The method at work in Theorems 1.1, 1.2 and 1.5 is the dispersion method, which was pioneered by Linnik [Lin63] and studied intensively in groundbreaking work of Bombieri, Fouvry, Friedlander and Iwaniec [Fou82, FI83, BFI86] on primes in arithmetic progressions. It has received a large publicity recently with the breakthrough of Zhang [Zha14] (see also [PCF+14]), giving the first proof of the existence of infinitely many bounded gaps between primes (which was shown later by Maynard [May15] and Tao (unpublished) not to require such strong results). In our case, by writing τ(n) as a convolution of the constant function 1 with itself, the problem is reduced to estimating the mean value of Λ(n) or τk(n) when n x runs over arithmetic progressions (mod q), with an average over q. It is crucial that≤ the uniformity be good enough to average over q √x. In the case of Λ(n), that is beyond what can currently be done for individual≤ moduli q, even assuming the GRH. The celebrated theorem of Bombieri-Vinogradov [IK04, Theorem 17.1] allows to exploit

2 1/2 In [Mot76, FT85], the authors study the distribution of τk(n) in progressions of moduli up to x , while in [BV87] the authors address the distribution of τ(n) in progressions of moduli up to x1−1/k. ERROR TERM IN THE DISPERSION METHOD 5 the averaging over q, but if one wants error terms at least as good as O(x/(log x)2) for instance, it barely fails to be useful. Linnik’s dispersion method [Lin63], which corresponds at a technical level to an acute use of the Cauchy–Schwarz inequality, offers the possibility for such results, on the condition that one has good bounds on some types of exponential sums related to Kloosterman sums. One then appeals to Weil’s bound [Wei48], or to the more specific but stronger bounds of Deshouillers-Iwaniec [DI82b] which originate from the theory of modular forms through Kuznetsov’s formula. The Deshouillers-Iwaniec bounds apply to exponential sums of the following kind: rd bn,r,sg(c,d)e n c,d,n,r,s sc (rd,scX)=1 where c,d,n,r,s are integers in specific intervals, (bn,r,s) is a generic sequence, and g(c,d) depends in a smooth way on c and d. Here and in what follows, e(x) stand for e2πix, and rd denotes the multiplicative inverse of rd (mod sc) (since e(x) is of period 1, the above is well-defined). It is crucial that the variables c and d are attached to a smooth weight g(c,d): for the variable d, in order to reduce to complete Kloosterman sums (mod sc); and for the variable c, because the object that arises naturally in the context of modular forms is the average of Kloosterman sums over moduli (with smooth weight). In the dispersion method, dealing with largest common divisors (appearing through the Cauchy–Schwarz inequality) causes some issues. The most important of these is that the phase function that arises in the course of the argument takes a form similar to rd cd (2.1) e n + sc q rather than the above. Here q can be considered small and fixed, but even then, the second term oscillates chaotically. Previous works avoided the issue altogether by using a sieve beforehand in order to reduce to the favourable case q = 1. Two error terms are then produced, which take the form δ(log x)/ log z 1 e− + z− where z x is a parameter. Roughly speaking, the first term corresponds to sieving out prime≤ factors smaller than z, with the consequence that the “bad” variable q above is either 1 or larger than z. The second term corresponds to a trivial bound on the δ√log x contribution of q > z. The best error term one can achieve in this way is e− , whence the estimate (1.3). By contrast, in the present paper, we transpose the work of Deshouillers-Iwaniec in a slightly more general context, which allows to encode phases of the kind (2.1). More specifically, whereas Deshouillers and Iwaniec worked with modular forms with trivial multiplier system, we find that working with multiplier systems defined by Dirichlet characters allows one to encode congruence conditions (mod q) on the “smooth” variables c and d. This is partly inspired by recent work of Blomer and Milićević [BM15a]. The main result, which extends [DI82b, Theorem 12] and has potential for applications beyond the scope of the present paper, is the following.

Theorem 2.1. Let C,D,N,R,S 1, and q,c0,d0 N be given with (c0d0, q)=1. Let (b ) be a sequence supported inside≥ (R, 2R] (S,∈2S] (0, N] N3. Let g : R5 n,r,s × × ∩ + → 6 SARY DRAPPEAU

3 C be a smooth function compactly supported in ]C, 2C] ]D, 2C] (R+∗ ) , satisfying the bound × ×

ν1+ν2+ν3+ν4+ν5 ∂ g ν1 ν2 ν3 ν4 ν5 1 ε0 (2.2) (c,d,n,r,s) ν1,ν2,ν3,ν4,ν5 c− d− n− r− s− − ∂cν1 ∂dν2 ∂nν3 ∂rν4 ∂sν5 ≪ { } for some small ε > 0 and all fixed ν 0. Then 0 j ≥ rd bn,r,sg(c,d,n,r,s)e n c n r s sc X Xd X X X (2.3) c c0 and d d0 (mod q) ≡ (qrd,sc≡ )=1 (CDNRS)ε+O(ε0)qK(C,D,N,R,S) b , ≪ε,ε0 k N,R,Sk2 1/2 where b = b 2 and k N,R,Sk2 n,r,s | n,r,s| P 2 2 2 1 K(C,D,N,R,S) = CS(RS + N)(C + RD)+ C DS (RS + N)R + D NRS− . q We have made no attempt to optimize the dependence in q. In all of the applications considered here, we only apply the estimate (2.3) for small values of q, ε1 say q = O((CDNRS) ) for some small ε1 > 0. Such being the case, the reader might still wonder why the bound tends to grow with q. The main reason is that upon com- pleting the sum over d, we obtain a Kloosterman sum to modulus scq, which grows with q. In the footsteps of previous work [Dra15], for the proof of our equidistribution results, we separate from the outset of the argument the contribution of characters of small conductors (which is typically well-handled by complex-analytic methods). We only apply the dispersion method to the contribution of characters of large conductors. There is considerable simplification coming from the fact that no “Siegel-Walfisz”-type hypothesis is involved in the latter, which allows us to focus on the combinatorial aspect of the method3. In Section 3, we state a few useful lemmas. In Section 4, we adapt the arguments of [DI82b] to prove Theorem 2.1. In Section 5, we employ a variant of the dispersion method to obtain equidistribution for binary convolutions in arithmetic progressions. In Sections 6 and 7, we derive Theorems 1.1, 1.2, 1.5 and 1.6.

Notations. We use the convention that the letter ε denotes a positive number that can be chosen arbitrarily small and whose value may change at each occurence. The letter δ > 0 will denote a positive number whose value may change from line to line, and whose dependence on various parameters will be made clear by the context. The Fourier transform f of a function f is by deﬁnition

b f(ξ)= f(t)e(ξt)dt. R Z If f is smooth and compactly supported,b the above is well-deﬁned and there holds

f(t)= f(ξ)e( ξt)dξ. R Z − b 3 It is more straightforward to study the mean value of τk(n) in arithmetic progressions of small moduli, than a k-fold convolution of slowly oscillating sequences, each supported on a dyadic interval. ERROR TERM IN THE DISPERSION METHOD 7

(j) j If moreover f is supported inside [ M, M] for some M 1 and f M − for j 0, 2 , then we have − ≥ k k∞ ≪ ∈{ } M f(ξ) . ≪ 1+(Mξ)2 b 3. Lemmas In this section we group a few useful lemmas. The ﬁrst is the Poisson summation formula, which is very eﬀective at estimating the mean value of a smooth function along arithmetic progressions. Lemma 3.1 ([BFI86, Lemma 2]). Let M 1 and f : R C be a smooth function ≥ (j) → j supported on an interval [ M, M] satisfying f j M − for all j 0. For all q 1 and (a, q)=1, with− H := q1+ε/M, wek havek∞ ≪ ≥ ≥ 1 h ah 1 f(m)= f e − + O . q q q ε q m a X(mod q) hXH ≡ | |≤ b The next lemma is a very useful theorem of Shiu [Shi80, Theorem 2] and gives an upper bound of the right order of magnitude for sums of τk(n) in short intervals and arithmetic progressions of large moduli. It is an analog of the celebrated Brun- Titchmarsh inequality [IK04, Theorem 6.6]. We quote a special case. Lemma 3.2 ([Shi80, Theorem 2]). For k 2, x 2, x1/2 y x, (q, a) N with (a, q)=1 and q x3/4, ≥ ≥ ≤ ≤ ∈ ≤ y ϕ(q) k 1 τ (n) log x − . k ≪k q q x y

We quote from [Har11, Number Theory Result 1] the following version of the Pólya- Vinogradov inequality with an explicit dependence on the conductor. Lemma 3.4. Let χ (mod q) be a character of conductor 1 = r q, and M, N 1. Then 6 | ≥ χ(n) τ(q/r)√r log r. M

4.1. Setting. 4.1.1. Kloosterman sums. Let q 1. The setting is the congruence subgroup ≥ a b Γ=Γ (q) := SL (Z),c 0 (mod q) . 0 c d 2 ! ∈ ≡ κ Let χ be a character modulo q0 q, and κ 0, 1 such that χ( 1)=( 1) . We warn the reader that the variable q has| a different∈{ meaning} in Sections− 4.1 and− 4.2, than in the statement of Theorem 2.1 (where it corresponds to qrs). The character χ induces a multiplier (i.e. here, a multiplicative function) on Γ by a b χ = χ(d). c d ! The cusps of Γ are Γ-equivalence classes of elements R that are parabolic, i.e. each of them is the unique fixed point of some element of∪ Γ. {∞} They correspond to cusps on a fundamental domain. A set of representatives is given by rational numbers u/w where 1 w, w q, (u,w) = 1 and u is determined (mod (w,q/w)). ≤ | For each cusp a, let Γa denote the stabilizer of a for the action of Γ. A scaling matrix is an element σa SL (R) such that σa = a and ∈ 2 ∞ 1 b 1 σa σ− , b Z =Γa. 0 1 a ! ∈ Whenever a = u/w with u =0, (u,w) = 1 and w q, one can choose 6 | a [q,w2] 0 (4.1) σa = .  q 2 2 1 [q,w ] (a [q,w ])−  q q  A cusp a is said to be singular if χ(γ) = 1 for any γ Γa. When a = u/w with u and w as above, then this merely means that χ has conductor∈ dividing q/(w,q/w). The point at infinity is always a singular cusp, with stabilizer 1 Γ = ∗ . ∞ 0 1 ! For any pair of singular cusps a, b and any associated scaling matrices σa, σb, define the set of moduli

a b 1 (a, b) := c R∗ : a, b, d R, σ− Γσb . + c d a C ∈ ∃ ∈ ! ∈ This set actually only depends on a and b. For all c (a, b), let ab(c) be the set of real numbers d with 0

Moreover, the converse fact holds, that for any reals t1, t2, any cusps a and b, and any scaling matrices σa and σb, there exist scaling matricese e σa and σb associated to a and b such that the equality above holds. This rather simple fact is of tremendous help because all of the results obtained through the Kuznetsov fofrmulaf are uniform with respect to the scaling matrices, so that one can encode oscillating factors depending of m and n at no cost (it is crucial for separation of variables). Whenever the context is clear enough, we write Sab(m, n; c) without reference to the scaling matrices.

The ﬁrst example is a = b = and σa = σb = 1. Then ( , )= qN and ∞ C ∞ ∞ dm + dn (4.2) S (m, n; c)= χ(d)e (c qN) ∞∞ × c ∈ d (modX c) is the usual (twisted) Kloosterman sum. Here and in the rest of the paper, we write (mod c)× to mean a primitive residue class (mod c). The next example that we need is the case a = b. The following is an extension of [DI82b, Lemma 2.5]. It is proven in an identical way, so we omit the details. Lemma 4.1. Assume a = u/w is a cusp with (u,w)=1, w q and u = 0. Assume that a is singular. Choose the scaling matrix as in (4.1). Then | (a, a)= 6 q N, and C (w,q/w) if c = γq/(w,q/w) for some γ N, ∈ m n αδ 1 mα + nδ (4.3) Saa(m, n; c) = e (w,q/w) − ∗ χ α + u − e , uq γ c δ (modX c) where, in the sum ∗, δ runs over the solutions (mod c) of (4.4) (δ,γq/w)=1P , (γ + uδ,w)=1, δ(γ + uδ) u (mod (w,q/w)), ≡ and α is determined (mod c) by the equations

(4.5) αδ 1 (mod γq/w), α γ′u + u′(γ + u δ) (mod wγ′) ≡ ≡ ′ ′ ′ where γ′ = γ/(γ,u) and u′ = u/(γ,u).

The sums Saa(m, n; c) are expressed by means of the Chinese remainder theorem (twisted multiplicativity) as a product of similar sums for moduli c that are prime powers. When c = pν and ν 2, a bound is obtained by means of elementary methods as in [IK04, Section 12.3].≥ When c is prime, the Weil bound (cf. [KL13, Theorem 9.3]) from algebraic geometry can be used. In the general case, one obtains Lemma 4.2. For all c (a, a), m, n Z, we have ∈ C ∈ 1/2 O(1) 1/2 Saa(m, n; c) (m, n, c) τ(c) (cq ) ≪ 0 where q0 is the modulus of χ. Finally, we consider as in [DI82b] the following family of Kloosterman sums, which will be of particular interest to us. 10 SARY DRAPPEAU

Lemma 4.3. Assume that the level q is of the shape rs, with q0 r, where q0 is the modulus of χ, and (r, s)=1. The two cusps and 1/s are singular.| Choose the scaling matrices ∞ √r 0 σ = Id, σ1/s = . ∞ s√r 1/√r! Then ( , 1/s)= cs√r,c N, (c,r)=1 , and for (c,r)=1, we have C ∞ { ∈ } ns S ,1/s(m, n; cs√r)= χ(c)e S(mr, n; sc) ∞ r where S(. . .) in the right-hand side is the usual (untwisted) Kloosterman sum. The main feature here is the presence of the character outside the Kloosterman sums, as opposed to (4.2). It is proven in a way identical to [DI82b, page 240], keeping track of an additional factor χ(D) in the summand.

4.1.2. Normalization. In order to state the Kuznetsov formula, we first fix the normalization. We largely borrow from [BHM07a]. We also refer to [DFI02, Section 4] for useful explanations on Maaß forms, and to [Pro03] for a discussion in the case of general multiplier systems. For each integer k > 0 with k κ (mod 2), we fix a basis k(q, χ) of holomorphic cusp forms. It is taken orthonormal≡ with respect to the weighB t k Petersson inner product: k dxdy f,g k = y f(z)g(z) (z = x + iy). h i Γ H y2 Z \ We let (q, χ) denote a basis of the space of Maaß cusp forms. In particular they are functionsB on H, are automorphic of weight κ 0, 1 (meaning they satisfy [Pro03, formula (5)]), are square-integrable on a fundamental∈ { } domain and vanish at the cusps (note that when κ = 1, they do not induce a function on Γ H). They are eigenfunctions of the L2-extension of the Laplace-Beltrami operator \ 2 2 2 ∂ ∂ ∂ ∆= y 2 + 2 iκy . ∂x ∂y − ∂x This operator has pure point spectrum on the L2-space of cusp forms. For f (q, χ), 1 ∈ B we write (∆+ s(1 s))f = 0 with s = 2 + itf and tf R [ i/2, i/2]. The (tf )f (q,χ) form a countable− sequence with no limit point in ∈C (in∪ − particular, there are∈B only finitely many tf iR). We choose the basis (q, χ) orthonormal with respect to the weight zero Petersson∈ inner product. Let B

(4.6) θ := sup Im tf , f (q,χ) | | ∈B then Selberg’s eigenvalue conjecture is that θ = 0 i.e. tf R for all f (q, χ). Selberg proved that θ 1/4 (see [DI82b, Theorem 4]), and∈ the current best∈ B known result is θ 7/64, due to≤ Kim and Sarnak [Kim03] (see [Sar95] for useful explanations on this topic).≤ The decomposition of the space of square-integrable, weight κ automorphic forms on H with respect to eigenspaces of the Laplacian contains the Eisenstein spectrum (q, χ) which turns out to be the orthogonal complement to the space of Maaß E 1 a forms. It can be described explicitely by means of the Eisenstein series E (z; 2 + it) where a runs through singular cusps, and t R. Care must be taken because these are not square-integrable; see [IK04, Section∈ 15.4] for more explanations. ERROR TERM IN THE DISPERSION METHOD 11

Let j(g, z) := cz + d where g = ( ∗ ∗ ) SL (R). We write the Fourier expansion c d ∈ 2 of f (q, χ) around a singular cusp a with associated scaling matrix σa as ∈ Bk k k/2 (4.7) f(σaz)j(σa, z)− = ρfa(n)(4πn) e(nz). n 1 X≥ We write the Fourier expansion of f (q, χ) around the cusp a as ∈ B iκ arg j(σa,z) f(σaz)e− = ρfa(n)W |n| κ (4π n y)e(nx) n 2 ,itf n=0 | | X6 where the Whittaker function is deﬁned as in [Iwa02, formula (1.26)]. Finally, for every singular cusp c, we write the Fourier expansion around the cusp a of the Eisenstein series associated with the cusp c as

1 iκ arg j(σa,z) 1/2+it 1/2 it Ec(σaz, +it)e− = c1,c(t)y +c2,c(t)y − + ρca(n, t)W |n| κ (4π n y)e(nx). 2 n 2 ,it n=0 | | X6

4.1.3. The Kuznetsov formula. Let φ : R C be of class ∞ and satisfy + → C (j) 2 η (4.8) φ(0) = φ′(0) = 0, φ (x) (1 + x)− − (0 j 3) ≪ ≤ ≤ for some η > 0. In practice, the function φ will be ∞ with compact support in R+∗ . We deﬁne the integral transforms C

k ∞ dx (4.9) φ˙(k):= 4i Jk 1(x)φ(x) , 0 − x Z κ 2πit ∞ κ dx (4.10) φ(t) := (J2it(x) ( 1) J 2it(x))φ(x) , sinh(πt) Z0 − − − x e κ ∞ dx (4.11) φˇ(t):= 8i− cosh(πt) K2it(x)φ(x) Z0 x where we refer to [Iwa02, Appendix B.4] for the deﬁnitions and estimates on the Bessel functions. The sizes of these transforms is controlled by the following Lemma (we need only consider t 1/4 in the second estimate, by Selberg’s theorem that θ 1/4). | | ≤ ≤ Lemma 4.4 ([DI82b, Lemma 7.1], [BHM07b, Lemma 2.1]). If φ is supported on x X (j) j ≍ with φ X− for 0 j 4, then k k∞ ≪ ≤ ≤ φ(t) 1+ log X 1+ X 3 (4.12) φ˙(t) + | | + φˇ(t) | | min 1, (t R), | | 1+ t κ | | ≪ 1+ X 1+ t ∈ e | | | | 1+ X 2 t φ(t) + φˇ(t) − | | (t [ i/4, i/4]). | | | | ≪ 1+ X ∈ − e Proof. Taking into account the factor tκ in front of φ(t), the arguments of [DI82b, Lemma 7.1] and [BHM07b, Lemma 2.1] are easily adapted. The only non-trivial fact to e (j) j check is that the decaying factor in (4.12) only requires the hypotheses φ X− for j 4. This is seen by reproducing the proof of [BHM07b, Lemmak 2.1k]∞ with≪ the choices≤ j = 1 and i = 2.

Recall that κ is deﬁned by χ( 1) = ( 1)κ. We are ready to state the Kuznetsov formula for Dirichlet multiplier system− and− general cusps. 12 SARY DRAPPEAU

Lemma 4.5. Let a and b be two singular cusps with associated scaling matrices σa and σb, and φ : R C as in (4.8). Let m, n N. Then + → ∈ 1 4π√mn (4.13) Sab(m, n; c)φ = + + , c (a,b) c c H E M ∈CX 1 4π√mn (4.14) Sab(m, n; c)φ = ′ + ′, c (a,b) c − c E M ∈CX where , , (“holomorphic”, “Eisenstein”, “Maaß”) are deﬁned by H E M (4.15) := φ˙(k)Γ(k)√mnρ a(m)ρ b(n), H f f k>κ f k(q,χ) k κ X(mod 2) ∈BX ≡ 1 ∞ √mn (4.16) := φ(t) ρca(m, t)ρcb(n, t)dt, E c sing. 4π cosh(πt) X Z−∞ e√mn (4.17) := φ(tf ) ρfa(m)ρfb(n), M cosh(πtf ) f X(q,χ) ∈B e 1 ∞ √mn (4.18) ′ := φˇ(t) ρca(m, t)ρcb( n, t)dt, E c sing. 4π cosh(πt) − X Z−∞ √mn (4.19) ′ := φˇ(tf ) ρfa(m)ρfb( n). M f (q,χ) cosh(πtf ) − ∈BX Proof. For a = b = , the formula (4.13) and the case κ = 0 of (4.14) can be found in Section 2.1.4 of [BHM07a].∞ The extension to general cusps a, b is straightforward. The case κ = 1 of (4.14) was obtained by B. Topacogullari (private communications). The details are due to appear in forthcoming work, so we restrict here to mentionning that it can be proved by reproducing the computations of page 251 of [DI82b] and Section 5 of [DFI02]4. The right-hand side of the Kuznetsov formula (the so-called spectral side) naturally splits into two contributions. The regular spectrum consists in , and the contribution to of those f (q, χ) with t R ; the conjecturallyH E inexistant M ∈ B f ∈ exceptional spectrum is the contribution to of those f with t iR∗ (similarly M f ∈ with ′ and ′). The technical reason for this distinction is the growth properties of theE integralM transforms. Indeed, when X is small (i.e. when the average over the moduli of the Kloosterman sums is long, since X √mn/c), we see from Lemma 4.4 ≍ that while φ˙(t), φ(t) and φˇ(t) are essentially bounded for t R, φ(it) is roughly of 2 t ∈ size X− | | when t [ 1/2, 1/2]. We remark thate∈ in− contrast with other works (e.g. [BM15b]), wee do not make use of Atkin-Lehner’s newform theory, nor of Hecke theory. In fact, we do not use any information about the Fourier coeﬃcients ρfa(n) and ρca(n, t) other than the fact that Kuznetsov’s formula holds, so the reader unfamiliar with the subject can go through the following sections without knowing what they are. The main feature of the Kuznetsov formula which is used is the decay properties of the integral transforms (4.9)-(4.11), and the fact that it separates the variables m and n in a way that combines very nicely with the Cauchy–Schwarz inequality.

4Note that in the expression for h (t) given on page 518 of [DFI02], the term Γ(1 k ir) should p − 2 − read Γ(1 k + ir). − 2 ERROR TERM IN THE DISPERSION METHOD 13

4.2. Large sieve inequalities.

4.2.1. Quadratic forms with Saa. Given N N, ϑ R+∗ , λ 0, a sequence (bn) of complex numbers, a singular cusp a and c ∈ (a, a),∈ let ≥ ∈ C λ√mn 2√mn Ba(λ, ϑ; c, N) := bmbne− Saa(m, n, c)e ϑ . N

Proposition 4.7 ([DI82b, Proposition 4]). Let (an) be a sequence of complex numbers, and a a singular cusp for the group Γ0(q) and Dirichlet multiplier χ (mod q0). Suppose T 1 and N 1/2. Then each of the three quantities ≥ ≥ 2 (4.23) Γ(k) an√nρfa(n) , κ

κ 2 (1 + tf )± (4.24) | | an√nρfa( n) , f (q,χ) cosh(πtf ) N

4Γ(k 1)√mn ρ a(m)ρ a(n) − f f f Xk(q,χ) (4.26) ∈B k 1 4π√mn = 1m=n +2πi− Saa(m, n; c)Jk 1 , − c (a,a) c c ∈CX valid for k > 1, k κ (mod 2), and a “pre-Kuznetsov” formula [DFI02, Proposi- tion 5.2] which, for general≡ cusps, is (4.27) κ 2 Γ(1 2 + ir) 1 4π√mn ∓ | 1m=n + Saa( m, n; c)I 2 ± 4π c (a,a) c ± ± c ∈CX √mn 1 ∞ √mn = H(tf ,r)ρfa( m)ρfa( n)+ H(t, r)ρca( m)ρca( n) cosh(πtf ) ± ± 4π c cosh(πt) ± ± f (q,χ) sing. Z−∞ ∈BX X for all real r and positive integers m, n. Here, i κ 1 I (x)= 2x ( iv)± − K2ir(vx)dv (x> 0). ± − i − Z− where v varies on the half-circle v = 1, Re(v) 0 counter-clockwise. Note that by the complement formula | | ≥

2 π 1, ǫ =1, (4.28) Γ(1 ǫ + ir) = − 2 cosh(πr) ×  1 + r2, ǫ = 1.  4 −

Given the formulas (4.26) and (4.27), the arguments in [DI82b, pages 258-261] are adapted as follows. When κ = 0, the details are strictly identical. Consider the (k 1)/T case κ = 1 of (4.23). We multiply both sides of (4.26) by (k 1)e− − a a and − m n sum over k, m and n. The analog of the function EK(x) deﬁned in [DI82b, page 258] is (up to a constant factor) the function 1 2 ℓ 2ℓ/T 1 u xJ1(ux)du ET (x)= ( 1) 2ℓe− J2ℓ(x)= sinh 2 2 3/2 , ℓ 1 − − T Z0 (cosh(1/T ) u ) X≥ − as can be seen by reproducing the computations in [Iwa82, page 316]5. We then write 2 π/2 J1(y)= cos τ sin(y cos τ)dτ, π Z0 5 There is a slight convergence issue in the Fourier integral for yJ1(y), which is resolved by chang- ing b = cosh(1/T ) to b + iε, ε> 0 and letting ε 0. → ERROR TERM IN THE DISPERSION METHOD 15 split the integral at ∆ (0,π/2] and deduce the bound (4.23) by following the steps in [DI82b, page 259]. ∈ Consider next the case κ = 1 and positive sign of (4.24) and (4.25). We multiply 2 both sides of (4.27) by r cosh(πr)aman, integrate over r R and sum over m and n. The analog of the function Φ(x) of [DI82b, page 260] is the∈ function i ∞ 2 (r/T )2 Φ+(x)= r e− K2ir(xv)dvdr. i Z−∞ Z− y cosh ξ We use the expression K2ir(y)= 0∞ e− cos(2rξ)dξ (y > 0). For x> 0, we obtain by integrations by parts R 1 3 ∞ (ξT )2 1 Φ+(x)= i√πT e− ξ tanh ξ cos(x cosh ξ) cos(xϑ cosh ξ)dϑ dξ − 0 − 2 1 Z Z− 3 T ∞ (ξT )2 2 dξ = i√π e− (1 2(ξT ) ) sinh(x cosh ξ) , − x Z0 − cosh ξ and from there, the bounds (4.24) and (4.25) are obtained by reproducing the computations of [DI82b, page 261]. Consider ﬁnally the case of negative sign in (4.24) and (4.26). We multiply both 2 1 2 sides of (4.27) by r cosh(πr)/( 4 +r )aman. The analog of the function Φ(x) of [DI82b, page 260] is now i ∞ 2 (r/T )2 dv Φ (x)= r e− K2ir(xv) dr, − i v2 Z−∞ Z− and we have by integration by parts

i vx cosh ξ 3 ∞ (ξT )2 1 e− Φ (x)= i√πT e− ξ tanh ξ cos(ξ cosh ξ) dv dξ − 0 − 2i i v2 Z Z− 3 i xv cosh ξ T ∞ (ξT )2 2 1 e− dξ = i√π e− (1 2(ξT ) ) sinh(x cosh ξ)+ dv . − x 0 − i i v3 cosh ξ Z Z− From there, it is straightforward to reproduce the computations of [DI82b, page 261] using the bounds of Lemma 4.6.

4.2.3. Weighted large sieve inequalities for the exceptional spectrum. The objects we would like to bound now are of the shape 2 2 tf 1/2 Eq,a(Y, (an)) := Y | | ann ρfa(n) f (q,χ) N

Proof. The arguments in [DI82b, section 8.1, pages 270-271] transpose identically.6. The next step is to produce an analog of [DI82b, Theorem 6], which is concerned with the situation when an average over q is done. Deshouillers and Iwaniec make use of the very nice idea that with the choice a = for each q, the roles of q and c can be swapped in the Kuznetsov formula. Through an∞ induction process, this enhances signiﬁcantly the bounds obtained. This switching technique is speciﬁc to the choice a = for all q, with scaling matrices independent of q. ∞

Lemma 4.9. Assume the situation is as previously. Recall that χ has modulus q0 1. Then for all Y 1 and Q q , ≥ ≥ ≥ 0 ε 1 1/2 2 Eq, (Y, (an)) ε (QN) (Qq0− + N + NY ) aN 2, ∞ q Q ≪ k k X≤ q0 q | where the scaling matrices are chosen independently of q. Note that now, in the situation when N Q, the parameter Y is allowed to be as large as (Q/N)2 while still yielding a bound≤ of same quality as the regular spectrum. The ﬁnal situation is the special case when (an) is the characteristic sequence of an interval of integers. Then Deshouillers and Iwaniec are able to provide an even stronger bound [DI82b, Theorem 7], by enhancing the initial step in the induction. Lemma 4.10. Assume that the situation is as in Lemma 4.9. Assume moreover that (an)N

2 φ(tf ) 1/2 1 := g(q) ann ρf (n) . S cosh(πtf ) ∞ q 1 f (q0q,χ) n X≥ ∈BXR e X tf i ∈ This is seen by approximating the Bessel function in the deﬁnition of φ by its ﬁrst order term, as in [DI82b, formula (8.1)]. Opening the squares in 1 and applying the Kuznetsov formula and the large sieve estimates (Lemma 4.5 andS Propositione 4.7), one gets ε N 2 1 = aman 2(m, n)+ Oε (QNY ) Q + 1/2 an , S m,n S q n | | X 0 X g(q) 4π√mn 2(m, n) := φ S (m, n; qc), ∞∞ S q,c 1 q0qc q0qc X≥ Letting h(x) = h (x) = φ(x)g 4π√mn , one applies the Kuznetsov formula for m,n,c q0cx the group Γ0(q0c) (which requires that the scaling matrices be independent of q) and obtains ε N NY 2 1 3 + Oε (QNY ) Q + 1/2 + 1/2 an , S ≪ |S | q q Q n | | 0 0 X h(tf ) 3 := aman √mnρf (m)ρf (n). S m,n cosh(πtf ) ∞ ∞ C

R 1 ∞ q0c iτ ∞ iτ h(t)= g˘(iτ) κ,t(x)x φ(x)dxdτ. 2π 4π√mn 0 K Z−∞ Z Inserting into thee deﬁnition of and using the triangle inequality, we obtain S3 ∞ (1+iτ)/2 (1 iτ)/2 3 g˘(iτ) amm ρf (m) ann − ρf (m) S ≪ | | ∞ ∞ × Z−∞ C

From there, the arguments in [DI82b, page 273] apply and yiel d

∞ iτ 2 tf ε κ,t(x)x φ(x)dx ε Y | | + Y Z0 K ≪

from which the claimed bound follows in the same way as [DI82b, page 273].

Proof of (4.30). Assume that (an)N

1+ε 4π√mn 1 T := (q0QC)− φ S (m, n; kq0− ) . k ∞∞ k q0QC N

U1(M ′, N ′) := e((m n)ϑ)Sχ(m, n; k) . m M ′ − X≤ ′ n N ≤ Opening the summation in Sχ, we have δm δn U (k, M ′, N ′) U (k, M ′, N ′) := e + mϑ e nϑ . 1 2 k k ≤ δ (mod k)× m M ′ n N ′ − X X≤ X≤

It is crucial to note that the quantity on the RHS also exists for k not multiple of q0, so trivially 1+ε T (q0QC)− sup U2(k, M ′, N ′), ≪ N M ′,N ′ N 1 k q0QC ≤ ≤ ≍X From there on, the calculations in [DI82b, page 276] apply and yield, in the notation of [DI82b, Lemma 8.2],

U2(k, M ′, N ′) fM ′ (m)e(mϑ)fN ′ (n)e( nϑ)S(m, n; k). ≪ Z − m,nX ∈ b b The proof of Theorem 14 of [DI82b] follows through, and yields for all K 1, ≥ ε U2(k, M ′, N ′) ε (KMN) K(K + MN). k K ≪ X≤ Taking K q QC, we conclude that ≍ 0 T (q QC)ε(q QC + N 2). ≪ε 0 0 The rest of the arguments in [DI82b, page 277] applies and yields S(Q, N, Y, 0) (NY )ε(Q + N + Y )N ≪ε as claimed. ERROR TERM IN THE DISPERSION METHOD 19

Proof of Lemmas 4.9 and 4.10. In addition to the recurrence relation (4.29), we have the properties S(Q, Y, N, 0) (Y/Z)1/2S(Q, Z, N, 0) (1 Z Y ), ≤ ≤ ≤ N S(Q, 1, N, 0) (QN)ε Q + a 2. ≪ε 1/2 k N k2 q0 The second one follows from Proposition 4.7. Having these at hand, the induction arguments in [DI82b, page 274] and [DI82b, page 277] are easily reproduced. It is useful to notice that q0 appears only with negative powers in the error terms, and that its presence in the denominator of πNY/(q0Q) in (4.29) is beneficial for the induction. Remark. The previous three lemmas used only Selberg’s theorem that θ 1/4 (recall the definition (4.6)). One could make the bounds explicit in terms of θ and≤ thus benefit from recent progress towards the Ramanujan-Selberg conjecture. It is straightforward to check that Lemmas 4.8, 4.9 and 4.10 hold with the right-hand sides replaced by 2θ 1/2 1 2θ+ε 2 (1+(µ(a)NY ) )(1 + q (µ(a)N) − ) a , 0 k N k2 ε 1 2θ 4θ 1 4θ 2 (QN) (Qq− + N + Y N Q − ) a , 0 k N k2 ε 1 2θ 2θ 1 4θ (QN) (Qq0− + N + Y N Q − )N respectively (compare with [IK04, Proposition 16.10]). We refrain from doing so because it would not impact the applications considered here. 4.3. Proof of Theorem 2.1. 4.3.1. Estimates for sums of generalized Kloosterman sums. We begin by the following statement regarding the generalized Kloosterman sums Sa,b(m, n; c). For the sake of simplifying the presentation of the bound obtained, we discard powers of the modulus q. This does not have consequences on our applications. Proposition 4.12. Let the real numbers M,N,R,S 1, X > 0 and the integer q 1 be given, let χ be a character modulo q, let φ be a smooth≥ function supported on≥ the (j) j interval [X, 2X] such that φ X− for 0 j 4, and let (am) and (bn,r,s) be sequences of complex numbersk supportedk∞ ≪ on M <≤ m ≤2M, N < n 2N, R

Remark. If (am) is not the characteristic sequence of an interval, then the bound (4.31) 1/4 still holds with Lexc is replaced by M Lexc (see [DI82b, Theorems 10 and 11]). Proof. This estimate is deduced from Proposition 4.7 and Lemmas 4.8 and 4.10 by following the computations of Section 9.1 of [DI82b]. It is useful to notice that the bounds of Lemmas 4.8, 4.10 and Proposition 4.7 (for a , 1/s ) decrease with q0. ∈ {∞ }

4.3.2. Estimates for the complete Kloosterman sums twisted by a character. We now justify the transition from Proposition 4.12 to an estimate for twisted sums of usual Kloosterman sums S(m, n; c). Proposition 4.13. Let the real numbers M,N,R,S,C 1, and the integer q 1 be given, let χ be a character modulo q, let g be a smooth function≥ supported on [C,≥2C] 3 × [M, 2M] (R∗ ) such that × + ν0+ν1+ν2+ν3+ν4 ∂ g ν0 ν1 ν2 ν3 ν4 (4.32) (c, m, n, r, s) C− M − N − R− S− ∂cν0 ∂mν1 ∂nν2 ∂rν3 ∂sν4 ≪ for 0 νj 12. Let (bn,r,s) be a sequence of complex numbers supported on N < n 2N, R

4π√x1x2 4π√x1x2 g (x, x1,...,x4) := g , x1,...,x4 . ∗ x xx4√x3q By integration by parts, for any non-negative integers (ℓ,ℓ1,...,ℓ4) with ℓ 4 and ℓj 2, ≤ ≤

ℓ ℓ+ℓ1+ +ℓ4 ∂ G ℓj ∂ ··· (x, ξ1,...,ξ4)= (2πiξj)− g (x, x1,...,x4) ∂xℓ R4 ∂xℓ∂xℓ1 ∂xℓ4 ∗ × Yj Z 1 ··· 4 e(x ξ + + x ξ ) dx × 1 1 ··· 4 4 j Yj ERROR TERM IN THE DISPERSION METHOD 21 assuming ξj =0 if ℓj > 0. The derivatives are estimated using (4.32). Choose ℓ1 =0 or ℓ =2 according6 to whether ξ M < 1 or not, and similarly for ℓ , ℓ , ℓ . Then 1 | 1| 2 3 4 ℓ 2 ℓ ∂ G MNRS C√qR(√MN/(CS√qR))− ℓ (x, ξ1,...,ξ4) 2 2 2 2 . ∂x ≪ (1+(ξ1M) )(1+(ξ2N) )(1+(ξ3R) )(1+(ξ4S) ) We abbreviate further (1+(ξ M)2)(1+(ξ N)2)(1+(ξ R)2)(1+(ξ S)2) φ(x)= φ (x) := 1 2 3 4 G(x, ξ ,...,ξ ). ξ1,...,ξ4 MNRS2C√qR 1 4

This function satisﬁes the hypotheses of Proposition 4.12, with7 X = √MN/(CS√qR), uniformly in ξj. Deﬁne

b := b e(n(ξ + s/(rq)) rξ sξ . n,r,s n,r,s 2 − 3 − 4 e Finally, by Lemma 4.3 with an appropriate choice of scaling matrix (depending on ξ1 and t), we have

χ(c)S(nr, mq; sc)e(m(t ξ1)) = S ,1/s(m, n; sc√rq). − ∞ Proposition 4.12 can therefore be applied and yields 1 4π√mn bn,r,s φ S ,1/s(m, n; sc√rq) m,n,r,s cs√rq sc√rq ∞ X (c,rqX)=1 (s,rq)=1 e q3/2(CMNRS)ε (W + W )√M b , ≪ε CS√qR reg exc k N,R,Sk2 with (C2S2R + MN + C2SN)(C2S2R + MN + C2SM) W 2 = RS , reg C2S2R + MN 2 3 2 Wexc = C S R(N + RS). q From the deﬁnitions of φ and G, we deduce the claimed bound.

4.3.3. Bounds for incomplete Kloosterman sums. In this section, we prove Theorem 2.1. As a first reduction, we remark that it suffices to prove the result when the sequence bn,r,s is supported on N < n 2N, by summing dyadically over N and by 1≤/2 concavity of √ (losing a factor (log N) in the process). Secondly, we let s0 (mod q)× be fixed and assume· without loss of generality that

(4.34) b =0 unless s s (mod q). n,r,s ≡ 0 1/2 We will recover the full bound (2.3) by summing over s0 (mod q)× (losing a factor q in the process by concavity). Let

(4.35) g¨(c, m, n, r, s) := ∞ g(c,ξ,n,r,s)e(ξm)dξ. Z−∞

7Note that in [DI82b, page 278], some occurences of X should read X−1. 22 SARY DRAPPEAU

By Poisson summation, we write the left-hand side of (2.3) as rδ bn,r,s e n g(c,d,n,r,s) c,n,r,s δ (mod sc) sc d δ (mod sc) (qr,scX)=1 X ≡ X (δ,sc)=1 d d0 (mod q) c c0 (mod q) ≡ ≡ b rδ md sc mδq = n,r,s e n g¨(c,m/sqc,n,r,s)e 0 c,n,r,s scq (δ,sc)=1 sc m − q − sc (qr,scX)=1 X X c c0 (mod q) ≡ b md s c (4.36) = n,r,s g¨(c,m/scq,n,r,s)e − 0 0 0 S(nr, mq; sc) c,m,n,r,s scq q − (qr,scX)=1 c c0 (mod q) ≡ where S(. . .) is the usual Kloosterman. Let M > 0 be a parameter. We write (4.36) as 0 + + , where 0 is the contribution of m = 0, is the contribution of indicesA mAsuch∞ thatB m >A M, and is the contribution of indicesA∞ m with 0 < m M. By the bound for Ramanujan| | sumsB [IK04, formula (3.5)], | | ≤

1 bn,r,s 2 2 1/2 0 | | g¨(c, 0, n, r, s) (n,sc) q− (log S) D NR/S bN,R,S 2. A ≪ q c,n,r,s sc | | ≪ { } k k (qr,scX)=1 c c0 (mod q) ≡ By repeated integration by parts in the integral (4.35), for ﬁxed k 1 and m = 0 we have ≥ 6 k 1 k(1 ε0) scq g¨(c,m/(scq), n, r, s) D − − . ≪k m | | ε+O(ε0) Taking k 1/ε0, we have that there is a choice of M (SCqD) SCq/D such that the bound≍ ≪ g¨(c,m/(scq), n, r, s) 1/m2 ( m > M) ≪ε | | holds. Bounding trivially the Kloosterman sum in (4.36) by sc, we obtain

ε+O(ε0) 2 1/2 (4.37) ε (SCqD) q− D NR/S bN,R,S 2 A∞ ≪ { } k k 2+ε+O(ε0) which is also acceptable (if ε0 is small enough, the factor q− is bounded). There remains to bound ; we may assume that M 1 for otherwise is void. By dyadic decomposition, B ≥ B

log M sup (M1) , |B| ≪ 1/2 M1 M |B | ≤ ≤ where

bn,r,s md0s0c0 (M1) := g¨(c,m/scq,n,r,s)e − S(nr, mq; sc). B c,m,n,r,s scq q − (qr,scX)=1 M1< m 2M1 | |≤ c c0 (mod q) ≡ We insert the deﬁnition of g¨ after having changed variables ξ ξscq/m, to obtain → DM1 (M1) sup ′(M1,ξ) , |B |≪ SCq ξ DM1/(SQq) |B | ≍ ERROR TERM IN THE DISPERSION METHOD 23 where (4.38) bn,r,s md0s0c0 ′(M1,ξ) := g(c,ξscq/m,n,r,s)e − S(nr, mq; sc). B c,m,n,r,s m q − (qr,scX)=1 M1< m 2M1 | |≤ c c0 (mod q) ≡ By orthogonality of multiplicative characters, we have 1 ′(M1,ξ)= χ(c0) (M1,ξ,χ), B M1ϕ(q) S χ (modX q) where

md0s0c0 (M1,ξ,χ) := bn,r,s χ(c)g1(c, m, n, r, s)e − S(nr, mq; sc), S r,s m,n (c,rq)=1 q − (s,qrX)=1 mXM1 X | |≍

1 g1(c, m, n, r, s) := M1m− g(c,ξscq/m,n,r,s). Proposition 4.13 can be applied to the sums (M ,ξ,χ), at the cost of enlarging the S 1 bound by a factor O((CDNRS)60ε0 ) in order for the derivative conditions (4.32) to be satisﬁed. We obtain

(M ,ξ,χ) q3/2(CDNRS)ε+O(ε0) L + L M b , S 1 ≪ε reg exc 1k N,R,Sk2 n oq 2 2 2 2 2 2 2 (C S R + M1N + C SN)(C S R + M1N + C SM1) Lreg := RS 2 2 , C S R + M1N

2 3 2 Lexc := C S R(N + RS). q From there, computations identical to [DI82b, page 282] allow to bound

C2M N L RS C2S2R + M N + 1 + C2S(M + N) . reg ≪ 1 R 1 We deduce successively

1/2 √ ε+O(ε0) q D M1 (M ) (CDNRS) L∗(M ) b , |B 1 |≪ε SC 1 k N,R,Sk2

2 2 2 2 2 3 2 L∗(M1) := RS(C S R + M1N + C M1N/R + C S(M1 + N)) + C S R(N + RS), q and ﬁnally

(4.39) (CDNRS)ε+O(ε0)q1/2 , B≪ε K

2 := CS(N + RS)(C + RD)+ C2DS (N + RS)R. K q Grouping our two bounds (4.37) and (4.39), and summing over s0 (mod q)×, we obtain the claimed result. 24 SARY DRAPPEAU

5. Convolutions in arithmetic progressions In this section, we proceed with an instance of the dispersion method, for convolu- η 1/3 η tions of two sequences one of which is supported in [x , x − ] for some η > 0. This extends [BFI86, Section 13] and [Fou85, Section V]. Given a parameter R 1, an integer q 1 and a residue class n (mod q), we let ≥ ≥ (R) := χ (mod q), cond(χ) R , Xq { ≤ } and 1 uR(n; q) := 1n 1 (mod q) χ(n) ≡ − ϕ(q) χ q(R) ∈XX (5.1) 1 = χ(n). ϕ(q) χ (modX q) cond(χ)>R Note that this vanishes when q R. We have the trivial bound ≤ Rτ(q) (5.2) uR(n; q) 1n 1 (mod q) + . | |≪ ≡ ϕ(q) It will also be sometimes useful to write

1(n,q)=1 1 (5.3) uR(n; q)= 1n 1 (mod q) χ(n). ≡ − ϕ(q) − ϕ(q) χ (modX q) 1

O(1) 1 (5.6) αmβnuR(mna1a2; q) x(log x) R− . Q

Introducing uR(n; q) is technically much more convenient than the usual

1(n,q)=1 (5.7) u1(n; q)= 1n 1 (mod q) . ≡ − ϕ(q) Indeed, there are no equidistribution assumptions on our sequences in Theorem 5.1. ERROR TERM IN THE DISPERSION METHOD 25

5.1. Bombieri-Vinogradov range. Before we embark on the dispersion method we 1/2 ε need an estimate which is relevant to values of the moduli less than the threshold x − . Lemma 5.2. Let M,N,R 1. Let x = MN, and suppose we are given two se- ≥ quences (αm) and (βn) supported on the integers of (M, 2M] and (N, 2N] respectively, satisfying the bounds (5.4). Suppose that Q x1/2/R and R Q. Then ≤ ≤ O(1) 1 1/2 1/2 max αmβnuR(mna; q) x(log x) (R− + M − + N − ). 0

Proof. See [IK04, Theorem 17.4]. Only the case r > R appears in our case. 5.2. First reductions. First we apply two reductions, following Section V.2 of [Fou85] and Section 3 of [FI83]. We replace the sharp cutoﬀ for the sum over q by a smooth function γ(q) ; and we transfer the squareful part of n into the number a2, allowing us to assume that n is squarefree. Note also that the statement of Theorem 5.1 is monotonically weaker as δ 0, so that whenever needed, we will take the liberty of reducing the value of δ in a→ way that depends at most on η.

Proposition 5.3. Let x,M,N,Q,R,η and the sequences (αm) and (βn) be as in The- orem 5.1. Assume that (βn) is supported on squarefree integers. There exists δ > 0 such that for any smooth function γ : R [0, 1] with + → (5.8) 1q (Q,2Q) γ(q) 1q (Q/2,3Q/2], ∈ ≤ ≤ ∈ (j) j+Bδj and γ j Q− for some B 0 and all ﬁxed j 0, under the condi- tionsk(5.5)k,∞ we≪ have ≥ ≥ O(1) 1 (5.9) γ(q) αmβnuR(mna1a2; q) x(log x) R− . q n,m ≪ (q,aX1a2)=1 (n,aX2)=1 The implicit constants depend on η, A (in (5.4)), B and the function γ at most. Proof that Proposition 5.3 implies Theorem 5.1. We replace the sharp cutoﬀ Q < q 2Q by a smooth weight γ(q) such that ≤

1q (Q,2Q] γ(q) 1q (Q(1 Q−10δ ),2Q(1+Q−10δ )]. ∈ ≤ ≤ ∈ − (j) j+10δj We can pick γ such that γ j Q− for all ﬁxed j 0. The error term in this procedure comes fromk k the∞ ≪ contribution of those integers≥ q at the transition 10δ 10δ range 2Q < q 2Q(1 + Q− ) and Q(1 Q− ) q Q. It is bounded by the triangle inequality,≤ using our trivial bound− (5.2)≤ and≤ following the reasonning 10δ of [BFI86, page 219 and 240], choosing Q0 = x there. We obtain O(1) 10δ (5.10) (1Q

1 1/2 K− (K 1). k K k ≪ ≥ kX≥ ∈K 26 SARY DRAPPEAU

Proceeding as in [Fou85, Section V.2] and using the trivial bound (5.2), we deduce for any K 1, ≥ γ(q) αmβnuR(mna1a2; q) q n,m (q,aX1a2)=1 (n,aX2)=1 2 u (5.11) = γ(q) αmµ(n) βkn R(mnka1a2; q) k K q n,m kX≤ (q,aX1a2)=1 (n,kaX2)=1 (k,a∈K2)=1 O(1) 1/2 + O(Rx(log x) K− ). We are left to analyze, for k , k K, (k, a )=1, the sum ∈K ≤ 2 2 γ(q) αmβknµ(n) uR(mna1ka2; q). q n,m (q,aX1a2)=1 (n,kaX2)=1

4δ δ 2 Assume K x . For each ﬁxed k, the sequences (αm)m and (k− µ(n) βkn)n are supported in m≤ (M, 2M] and n (N/k, 2N/k], respectively. We apply Proposition 5.3 ∈ ∈ δ with η replaced by η/2, N replaced by N/k and a2 replaced by ka2 (the factor k− δ 2 ensures that the condition (5.4) holds for (k− µ(n) βkn)n). If δ is small enough in terms of η, we obtain, uniformly for k K, ≤ 2 1+δ O(1) 1 γ(q) αmβknµ(n) uR(mna1ka2; q) k− x(log x) R− . q n,m ≪ (q,aX1a2)=1 (n,kaX2)=1

1+δ Note that the sum k k− converges. Inserting in (5.11), we obtain ∈K P O(1) 1 1/2 γ(q) αmβnuR(mna1a2; q) x(log x) (R− + RK− ) q n,m ≪ (q,aX1a2)=1 (n,aX2)=1 and so we conclude by the choice K = R4.

5.3. Applying the dispersion method. Let us prove Proposition 5.3. Recall that the sequence (βn) is assumed to be supported on squarefree integers. Let denote the left-hand side of (5.9). By the triangle inequality D

= γ(q) αmβnuR(mna1a2; q) αm . |D| (q,a a )=1 m,n ≤ m | | q n X1 2 X X X X (n,a2)=1

Let the function α(m) be ∞ with α(m) 1 for M < m 2M, supported in- C (j) ≥ j ≤ side [M/2, 2M] and such that α j M − . Then by the Cauchy–Schwarz inequality and the hypothesis (5.4),k k∞ ≪

1/2 (5.12) (log x)O(1)M 1/2 2 Re + |D| ≪ S1 − S2 S3 where

1 = γ(q1)γ(q2) βn1 βn2 α(m) S n ,n (q1q2,a1a2)=1 1 2 mn1 a1a2 (mod q1) X (n1nX2,a2)=1 ≡ X mn2 a1a2 (mod q2) ≡ and and are deﬁned similarly, replacing the sum over m by S2 S3 1 χ(mna1a2) α(m)χ2(m), ϕ(q2) χ2 q (R) n1 a1a2 (mod q1) ∈XX2 ≡ X ERROR TERM IN THE DISPERSION METHOD 27

1 χ1(n1a1a2)χ2(n2a1a2) α(m)χ1χ2(m) ϕ(q1)ϕ(q2) χ1 q (R) χ2 q (R) (mn1,q1)=1 ∈XX1 ∈XX2 X (mn2,q2)=1 respectively. We will prove O(1) 2 2 (5.13) 2 Re + = O((log x) MN R− ). S1 − S2 S3 5.3.1. Evaluation of 3. The term 3 is deﬁned by (5.14) S S γ(q1)γ(q2) 3 = βn1 βn2 α(m)χ1(mn1a1a2)χ2(mn2a1a2). S ϕ(q1)ϕ(q2) n1,n2 (q1q2,a1a2)=1 χ1 q1 (R) (m,q1q2)=1 X ∈XX (nj ,qXja2)=1 X χ2 q (R) ∈X 2 1+ε Let W := [q1, q2] and H := W /M. By Poisson summation (Lemma 3.1), α(0) α(m)χ1χ2(m) = χ1χ2(b) m W × X b b (modXW ) 1 h bh 1 + α e − χ χ (b)+ O . W W W 1 2 ε W 0< h H b (mod W )× X| |≤ X b 8 The conductor of χ1χ2 is at most R, so that [IK04, Lemma 3.2] yields bh e − χ χ (b) R1/2 d. W 1 2 b (mod W )× ≪ d (h,W ) X |X We deduce

α(0) ε 1/2 α(m)χ1χ2(m)= χ1χ2(b)+ Oε(W R ). m W × X b b (modXW ) The error term is O(xδ) while the trivial bound is M x2/3. We deduce ≥ 2 1/2 = α(0)X + O(MN x− ), S3 3 where, having changed b to ba a , 1 2b γ(q1)γ(q2) X3 := βn1 βn2 χ1(bn1)χ2(bn2). q1,q2 [q1, q2]ϕ(q1)ϕ(q2) n1,n2 × χ1 q1 (R) b (mod W ) (q1q2,aX1a2)=1 ∈XX (nj ,qXj a2)=1 X χ2 q (R) ∈X 2 By orthogonality,

χ1χ2(b)= ϕ(W )1χ1 χ2 × ∼ b (modXW ) where by χ χ we mean that χ and χ are induced by the same primitive character 1 ∼ 2 1 2 – which necessarily has conductor dividing (q1, q2). Therefore,

χ1(n1)χ2(n2)1χ1 χ2 = χ0(n1n2). ∼ χ1 q (R) χ0 (R) ∈XX1 ∈X(Xq1,q2) χ2 q (R) ∈X 2 Since ϕ([q1, q2]) = ϕ(q1)ϕ(q2)/ϕ((q1, q2)), we deduce

γ(q1)γ(q2) (5.15) X3 = βn1 βn2 χ0(n1n2). (q q ,a a )=1 [q1, q2]ϕ((q1, q2)) n1,n2 1 2 1 2 χ0 (q1,q2)(R) X ∈XX (nj ,qXj a2)=1

8Note that in Lemma 3.2 of [IK04], τ(χ) should read τ(χ∗) and an additional factor χ∗(m/(dm∗)) should appear in the summand. 28 SARY DRAPPEAU

5.3.2. Evaluation of 2. The term 2 is deﬁned by (5.16) S S γ(q1)γ(q2) 2 = βn1 βn2 α(m)χ2(mn2a1a2). S ϕ(q2) n1,n2 (q1q2,a1a2)=1 χ2 q2 (R) m a1a2n2 (mod q1) X (nj ,qXj a2)=1 ∈XX ≡ X 1+ε As before, let W = [q1, q2] and H = W /M. By Poisson summation, α(0) 1 (5.17) α(m)χ2(m)= χ2(b)+ Oε 2 + , W × R W m a1a2n1 (mod q1) b (mod W ) ≡ X b X b a1a2n1 (mod q1) ≡ where M bh (5.18) := χ (b)e − . 2 W 2 W R 0< h H b (mod W )× X| |≤ X b a1a2n1 (mod q1) ≡ We wish to express the sum over b as a complete sum over residues. We write W = [q1, q2]= q1′ q2′ , where (q2′ , q1)=1 and q1′ q1∞ (meaning that p q1′ p q1). Then (q1′ , q2′ )= 1. Let | | ⇒ |

ψ :(Z/q1′ Z) (Z/q2′ Z) (Z/W Z) × 1 −→ denote the canonical ring isomorphism (so ψ− is the projection map). Note that b χ (ψ(1, b )) 2 7→ 2 2 deﬁnes a character (mod q2′ ) of conductor at most R. Finally, we have 1 q q 1′ + 2′ (mod 1). W ≡ q2′ q1′ The sum over b in (5.18) is in absolute values at most

b2hq1′ (5.19) χ2(ψ(1, b2))e − ′ × ′ × q′ b (mod q ) b2 (mod q ) 2 1 X 1 X 2 b1 a1a2n1 (mod q1) ≡ since ψ(b , b ) b (mod q ), and by factoring 1 2 ≡ 1 1 χ2(ψ(b1, b2)) = χ2(ψ(b1, 1))χ(ψ(1, b2)).

The sum over b2 in (5.19) is a Gauss sum; by [IK04, Lemma 3.2],

b2hq1′ 1/2 (5.20) χ2(ψ(1, b2))e − R d. ′ × q′ ≤ ′ b2 (mod q ) 2 d (h,q ) X 2 |X2

Note that

ϕ(q1′ ) (5.21) 1= =(q , q∞) ϕ(q ) 2 1 b (mod q′ )× 1 1 X 1 b1 a1a2n1 (mod q1) ≡ ν ν which is a shorthand for p q2, p q1 p . Multiplying (5.20) with (5.21) and summing || | over h, we obtain Q ε 1/2 W τ(q )(q , q∞)R . R2 ≪ε 2 2 1 Inserting this estimate into (5.17) then (5.16), the error term contributes τ(q )(q , q ) R1/2N 2W ε 2 2 1∞ xδ/2+εN 2Q. ≪ε q ≪ q1,q2 Q 2 X≍ ERROR TERM IN THE DISPERSION METHOD 29

In the last inequality we used standard facts about the kernel function k(n)= p n p, | for which we refer to [dB62]. The error term above is acceptable, since Q δ/2 1/2+2δ 2/3 2δ 2 x Q x x − MR− ≤ ≤ ≤ if δ is small enough. We therefore have 2 2 = α(0)X + O(MN R− ) S2 2 with (having changed b into ba a n ) 1 2 b2 γ(q1)γ(q2) X2 = βn1 βn2 χ2(b). [q1, q2]ϕ(q2) n1,n2 × (q1q2,a1a2)=1 χ2 q2 b (mod W ) X (nj ,qXja2)=1 X∈X X b n1n2 (mod q1) ≡ Fix χ and let χ (mod q ) be the primitive character inducing χ . Using 2 ∈ Xq2 2 2 2 orthogonality of characters (mod (q1, q2)), the sum over b is e e ϕ(q ) 2 1 χ2(b)= q2 (q0,q1)χ2(n1n2) × ϕ((q1, q2)) | b (modXW ) b n1n2 (mod q1) ≡ e e where we used the fact that (n n , (q , q ))=1. Summing over χ , we obtain 1 2 1 2 2 ∈ Xq2 ϕ(q2) χ2(b)= χ0(n1n2), × ϕ((q1, q2)) χ2 q b (mod W ) χ0 (q ,q ) X∈X 2 X ∈XX1 2 b n1n2 (mod q1) ≡ and so X2 = X3. 5.4. Second reduction. We now wish to evaluate

1 := γ(q1)γ(q2) βn1 βn2 α(m). S n ,n (q1q2,a1a2)=1 1 2 m a1a2n1 (mod q1) X (n ,q1Xq2a2)=1 ≡ X j m a1a2n2 (mod q2) n1 n2 (mod (q1,q2)) ≡ ≡

The expected main term is α(0)X1, where

d γ(q1)γ(q2) (5.22) X1 := βn1 βn2 . [q1, q2] n ,n (q1q2,a1a2)=1 1 2 X (njX,qj)=1 n1 n2 (mod (q1,q2)) ≡ For all integers q , n with (n , q )=1, let (q , n ) denote the contribution to 0 0 0 0 S1 0 0 S1 of those integers satisfying (q1, q2)= q0 and (n1, n2)= n0. Then we have (q , n ) xε α(m) |S1 0 0 |≪ε q1,q2 Q/q0 n1,n2 N/n0 a2n0n2m a1 (mod q0q2) X≍ X≍ ≡X (q0q2,a2n0)=1 n1 n2 (mod q0) q1 ma2n0n1 a1 ≡ | − (n2,q0q2)=1

ε Q ε x α(m) 1ma2n0n1=a1 ≪ q0 q2 Q/n0 n1,n2 N/n0 ma2n0n2 a1 (mod q0q2) ≍X X≍ ≡X (q0q2,a2n0)=1 n1 n2 (mod q0) ≡ (n2,q0q2)=1

+ 1ma2n0n1=a1 τ( ma2n0n1 a1 ) 6 | − | 2 2 ε MN MN Q N ε x 2 2 + + . ≪ n0q0 n0q0 n0q0 30 SARY DRAPPEAU

Therefore, for some δ > 0 and all 1 K xδ, we have ≤ ≤ ε 2 1 (q , n ) x MN K− . |S1 0 0 |≪ε (q0,nX0)=1 max q0,n0 >K { } By choosing K appropriately, it will therefore suﬃce to show that 2 δ δ (q , n )= α(0)X (q , n )+ O(MN x− ) (q , n x ) S1 0 0 1 0 0 0 0 ≤ where X (q , n ) is the contribution to X of indices with (q , q )= q and (n , n )= 1 0 0 b 1 1 2 0 1 2 n0. 5.5. Evaluation of (q , n ). Let the integers q , n be coprime, at most xδ, such S1 0 0 0 0 that (q0, a1a2)=(n0, a2)=1. Let us rename q1 into q0q1 and q2 into q0q2, and similarly for n1 and n2. We wish to evaluate

1(q0, n0)= γ(q0q1)γ(q0q2) βn0n1 βn0n2 α(m). S q ,q n ,n 1 2 1 2 m a1a2n0nj (mod q0qj ) (q1q2,a1a2X)=(q1,q2)=1 (n0nj ,qX0qj a2)=1 ≡ X (n1,n2)=1 n1 n2 (mod q0) ≡ Using Poisson summation, we have (q , n )= α(0)X (q , n )+ + O (xε ) S1 0 0 1 0 0 R1 ε R2 1+ε 1 where, having put W = q0q1q2 anddH := W M − , 1 h hµ = γ(q q )γ(q q )β β α e − , 1 0 1 0 2 n0n1 n0n2 W W W R q1,q2 n1,n2 0< h H X X X| |≤ b 1 2 2 2 = q0N , R q ,q n ,n W ≪ X1 2 X1 2 the summation conditions on qj and nj are the same as in the deﬁnition of 1(q0, n0), and the residue class µ (mod W ) satisﬁes S µ a a n n (mod q q ) (j 1, 2 ). ≡ 1 2 0 j 0 j ∈{ } 2 δ We seek an error term O(MN x− ). The contribution of 2 is acceptable. We now focus on . Recall that β is non-zero onlyR when n is squarefree (so R1 n that (n0, n1)=1). We have the equality modulo 1

µ a1 n1 n2 q1a2n0n2 q0q1q2n1 + a1 − a1 (mod 1). q0q1q2 ≡ q0q1q2a2n0n1 q0 n1q2 − a2n0 Taking the exponential, we may approximate a a e 1 =1+ O | 1| . q q q a n n q q q a n n 0 1 2 2 0 1 0 1 2 2 0 1 Inserting in , the error term contributes a quantity R1 a q Q2 N 2 | 1| 0 a N ≪ a n Q2N q2 n ≪| 1| | 2| 0 0 0 which is clearly acceptable. We therefore evaluate

γ(q0q1)γ(q0q1) h n1 n2 q1a2n0n2 q0q1q2n1 1′ := βn0n1 βn0n2 α e a1h − +a1h . R q ,q ,n ,n q0q1q2 q0q1q2 − q0 n1q2 a2n0 1 X2 1 2 Now we insert the deﬁnition of α as b h α = q0q1q2 α(q0q1q2ξ)e(hξ)dξ, q q q b R 0 1 2 Z b ERROR TERM IN THE DISPERSION METHOD 31 we detect the condition (a1, q1q2)=1 by Möbius inversion, and we split the sums over q1, q2 into congruence classes modulo n0a2. We obtain

1′′ := γ(q0δ1q1)γ(q0δ2q2) βn0n1 βn0n2 R q1,q2 n1,n2 × (δ1q1X,δ2q2)=1 (n0nj ,qX0δj qj )=1 qj λj δj (mod n0a2) (n1,n2)=1 ≡ n1 n2 (mod q0) ≡ q0λ1λ2n1 n1 n2 a2n0n2δ1q1 α(ξq0δ1δ2q1q2)e ξh + a1h e a1h − . × 0< h H a2n0 − q0 n1δ2q2 X| |≤

We write 1′′ in the form (2.3), with (5.24) R n1 n2 c q2, d q1, n a1h − , r a2n0n2δ1, s n1δ2, q n0a2, ← ← ← − q0 ← ← ← taking the complex conjugate or not depending on the sign of a1h(n1 n2), and with the term −

γ(q0δ1q1)γ(q0δ2q2)α(ξq0δ1δ2q1q2) playing the role of the function g. The derivative conditions (2.2) are satisﬁed with ε0 = Bδ, by virtue of our hypothesis on γ. At this point, we are in a situation analogous to [BFI86, formula (13.2)]. Applying Theorem 2.1, and evaluating the terms as in [BFI86, page 241], we obtain

O(δ) 1/2 1/2 ′′ x , R1 ≪ A B where HN 2 is the contribution coming from b 2 in (2.3), and A≪ k N,R,Sk2 Q2N 2N(H + N)+ Q3N 2√H + N + Q2HN (QN)2 N(H + N)+ Q√H + N . B≪ ≪ { } We have H xO(δ)N, so that Q2N 2xO(δ)(N 2 + Q√N) (compare with [BFI86, formula (13.4)]).≪ Inserting in (5.23),B ≪ we obtain

O(δ) 2 1 3/2 1/2 3/4 η/2+O(δ) 2 ′ x MN (Q− N + Q− N ) x− MN R1 ≪ ≪ 2/3 η by the hypothesis N Q − . Taking δ suﬃciently small in terms of η, we have the ≤2 δ required bound O(MN x− ).

5.6. The main terms. The main terms X1 and X3 deﬁned in (5.22) and (5.15) are real numbers. They combine to form

γ(q1)γ(q2) u X1 X3 = βn1 βn2 R(n1n2;(q1, q2)). − [q1, q2] n ,n (q1q2,a1a2)=1 1 2 X (nj ,qXja2)=1

Notice the summands are zero unless (q1, q2) > R. We use Möbius inversion

1(nj ,qj )=1 = µ(dj) d (q ,n ) j |Xj j 32 SARY DRAPPEAU to detect the conditions (nj, qj)=1, in order to separate the sums over n1, n2 from those over q1, q2. We insert the deﬁnition of uR in the form 1 uR(n1n2; q0)= χ(n1)χ(n2). ϕ((q1, q2)) χ primitive cond(Xχ)>R cond(χ) (q1,q2) | We can assume (dj, cond(χ))= 1 because of the factors χ(nj). Quoting from [Ten95, Theorem I.5.4] the bound ϕ(q) q/ log log q, we obtain ≫ 1 2 X1 X3 (log log x) βdj nχ(n) . − ≪ q ,q q1q2 R

For all t > R, the multiplicative large sieve inequality (Lemma 3.3) and our hypothesis (5.4) yields 2 O(1) 2A 2 G(t) := βdnχ(n) (log x) τ(d) (t + N)N R

ψ(x; q, a) := Λ(n), ψq(x) := Λ(n), ψ(x, χ) := Λ(n)χ(n). n x n x n x n a X(mod≤ q) (n,qX≤)=1 X≤ ≡ Let us recall the following classical theorem of Page [IK04, Theorems 5.26, 5.28]. Lemma 6.1. There is an absolute constant b such that for all Q, T 2, the following ≥ holds. The function s q Q χ (mod q) L(s, χ) has at most one zero s = β satisfy- Re 7→ ≤ Im ing (s) > 1 b/ log(QTQ) andQ (s) T . If it exists, the zero β is real and it is the zero of a unique− function L(s,| χ) for| some ≤ primitive real character χ.

Given a large x, we shall say thate χ is x-exceptional if the above conditionse are √log x met with Q = T = e . For all q 1 for which q q, we let χq denote the character (mod q) induced by χ. ≥e | e e e ERROR TERM IN THE DISPERSION METHOD 33

6.1. Primes in arithmetic progressions. We deduce from the previous sections the following result about equidistribution of primes in arithmetic progressions. Theorem 6.2. Assume the GRH. For some δ > 0, all x 1, Q x1/2+δ and all integers 0 < a , a xδ, ≥ ≤ | 1| | 2| ≤ 1 1 δ ψ(x; q, a1a2) ψq(x) x − . q Q − ϕ(q) ≪ X≤ (q,a1a2)=1 Unconditionally, under the same assumptions, 1 ψq(x)+ q qχ(a2a1)ψ(x, χq) δ√log x ψ(x; q, a1a2) | xe− , q Q − ϕ(q) ≪ X≤ e e (q,a1a2)=1 e where the term ψ(x; χq) is to be taken into account only if the x-exceptional character χ exists. e e Using the Dirichlet hyperbola method (see in particular section VII of [Fou85]), it 1 ε follows that the same estimate holds on the condition q x − for any fixed ε> 0 (the implicit constants and δ may then depend on ε). Note≤ however that the symmetry 1/2 1/2 point is at q (x a2 ) , rather than x (so the flexibility of taking Q somewhat larger than x1≈/2 is| not| superfluous). We refer to [Fio12b] for more explanations on what happens when Q is very close to x. As mentioned in the introduction, the uniformity in a1 and a2 is an interesting question. At the present state of knowledge, bounds coming from the theory of automorphic forms are typically badly behaved in that aspect. By using a more refined form of the combinatorial decomposition (6.4), Friedlander and Granville [FG92] prove 1/4 ε that a1 x − is admissible for all ε > 0 (in the case a2 = 1), with a somewhat larger| error| ≤ term. For the application to the Titchmarsh divisor problem, the following slightly weaker statement suffices. Proposition 6.3. For some δ > 0 and all x 2, assuming the GRH, we have ≥ 2 2 ψq(x) ψq(q ) 1 δ (6.1) ψ(x; q, a) ψ(q ; q, a) − x − . − − ϕ(q) ≪ qX√x (q,a≤)=1 Unconditionally, (6.2) 2 2 2 ψq(x) ψq(q ) ψ(x; χq) ψ(q ; χq) ψ(x; q, a) ψ(q ; q, a) − 1q qχ(a) − − − ϕ(q) − | ϕ(q) qX√x (q,a≤)=1 e e e δ√log x xe− . ≪ We will focus here on proving Proposition 6.3 only, because the presentation is slightly simpler and addresses all the essential issues. Proof of Proposition 6.3. Let 1 R x1/10 be a parameter. Let ≤ ≤ 1 := Λ(n). S 2 qX√x q

By orthogonality of characters, 1 (6.3) 1 = χ(na)Λ(n) S ϕ(q) 2 qX√x χ (modX q) q

µ(m1) µ(mj)(log n1)uR(n1m1 njmja; q) ··· 1/4 ··· ··· Q 0 be small. The contribution of tuples such that i MiNi x − is trivially 1 η+ε ≤ 1 η bounded by Oε(x − ) using Lemma 3.2. Suppose then i MiNi > x − . For conve- Q 1 δ nience we rename x = i MiNi. Our objective bound for (6.4)Q is O(x − ) and we now 1/4+η have Mi x if η isQ small enough. Fix η ≤(0, 1/100]. At least one of the three following cases must hold: ∈ 1 (2j 1)η (a) there exists an index k such that Nk > x − − , 1/3 η (b) we have min Nk, Nk′ > x − for two indices k = k′, { } 6 η 1/3 η (c) there exists an index k such that Mk or Nk lies in the interval [x , x − ]. In case (a), our sum (6.4) is at most

ε (6.5) a := x βnuR(mna; q) S Q

1 7η with β = 1 or log, MN = x and N x − . Choose η < 1/30, for the sum over n, we ≥ express uR as (5.3). Using z (6.6) 1= + O(1) (z 1, (a, q) N2) n z q ≥ ∈ n a X(mod≤ q) ≡ and partial summation in case β = log, we get that the sum over n above is 1 β u (mna; q) log x + β χ(n) . n R ≪ ϕ(q) n (1 ∆)XN

Dropping the condition cond(χ) R, we obtain for (6.5) a crude bound ≤ := x xεMQR1/2 QR1/2x8η x11/20+8η+δ Sa ≪ε ≪ ≪ which is acceptable. Consider case (b). Then the sum on the LHS of (6.4) is of the form

(6.7) b := α(m)β(n)γℓuR(mnℓa; q) S Qx − , MNL = x, α and β are either 1 or log, and γℓ satisﬁes

γℓ τ2j 2(ℓ)log ℓ | | ≤ − By partial summation and upon rewriting the size restrictions on m, n, ℓ, q as diﬀer- ences of one-sided inequalities, it suﬃces to establish the bound

1 δ ′ := u (mnℓa; q) x − b R S ℓ L q Q m M n N ≪ X≤ X≤ X≤ X≤ (q,aℓ)=1

1/3 2η whenever M,N > x − and Q 2√x. Writing uR as in (5.3), we have by the triangle inequality ≤ ′ ′ + ′ , Sb ≪ Sb1 Sb2 where

′ = u (mnℓa; q) , b1 1 S ℓ L q Q m M n N X≤ X≤ X≤ X≤ (q,aℓ)=1

1 b′2 = χ(m) χ(n) . S ℓ L q Q ϕ(q) χ (mod q) m M n N X≤ X≤ X X≤ X≤ 1

(6.8) αmβnuR(mna; q) Q

We consider now −, which we recall is S1 1 1− = Λ(n)χ(na). S ϕ(q) 2 qX√x χ (modX q) q 0. Choose R = ec√log x/2. We write 2 1 2 ψq(x) ψq(q )+ q qχ(a)(ψ(x; χq) ψ(q ; χq)) ♭ c√log x/2 − = − | − + + O(xe− ), S1 ϕ(q) S1 qX√x e e (q,a≤)=1 e the error term being there to cover the trivial case when q > R (so χ was not counted in −). By the same computation as above, S1 ♭ c√log x c√loge x/3 e Rx(log x)e− xe− . S1 ≪ ≪ This concludes the proof of (6.2) hence of Proposition 6.3.

6.2. Proof of Theorems 1.1 and 1.2. It is now straightforward to deduce Theo- rems 1.1 and 1.2. By the Dirichlet hyperbola method [FT85, page 45], we have T (x)=2 ψ(x; q, 1) ψ(q2; q, 1) + O(√x). q √x − ≤X Assume ﬁrst the GRH. Then Proposition 6.3 yields 2 ψq(x) ψq(q ) 1 δ T (x)=2 − + O(x − ) q √x ϕ(q) ≤X The GRH [MV07, formula (13.19)] allows us to deduce 2 x q 1 δ T (x)=2 − + O(x − ). q √x ϕ(q) ≤X The main term is computed using [Fou82, Lemme 6], which yields the claimed estimate. ERROR TERM IN THE DISPERSION METHOD 37

Unconditionally, from Proposition 6.3, we merely have to add to our estimate for T (x) the additional contribution of the x-exceptional character (if it exists), which takes the form ψ(x; χ ) ψ(q2; χ ) (6.9) 2 q − q q √x ϕ(q) X≤ e e q q | We have from [MV07, Theorem 11.16] e β x δ√log x ψ(x; χ )= + O(xe− ) q − β and similarly e 2β 2 q δ√log x ψ(q ; χ )= + O(xe− ) q − β at the possible cost of changinge the numerical value of δ. We obtain that (6.9) equals

β 2β 2 x q δ√log x − + O(xe− ). −β q √x ϕ(q) ≤X q q | The sums over q are computed using [Fou82, Lemme 6] (and partial summation in the β 2β x β 1 e form x q = β 2 t − dt), which yields Theorem 1.2. Corollary 1.3 is straightfor- − q ward. R There remains to justify Corollary 1.4. Note that C2(q) is absolutely bounded, √log x β β β 1 while q e by deﬁnition. Therefore x , and β li(x )/x (log x)− . We deduce ≤ → ∞ e ∼ e log q + C (q) γ 2 − 0 β β x x /(β li(x )) −→→∞ e e in an eﬀective way. For x large enough, it is less than 1/3 and Corollary 1.4 follows. Remark. If we were to consider τ(n a) instead of τ(n 1), for some a which is not a perfect square, then the Siegel zero− contribution (if it− existed) would have a twist by χ(a), which is a priori of unpredictable sign.

7. Application to correlation of divisor functions In this section, we justify Theorem 1.5. The proof has the same structure as that of Theorems 1.1 and 1.2, replacing the function Λ(n) by τk(n).

7.1. An equidistribution estimate. The analog of Theorem 6.2 is the following: Theorem 7.1. There exists η > 0 such that under the conditions k 4, 0 < a xη and Q x1/2+η , ≥ | | ≤ ≤ 1 1 η/k (7.1) τk(n) τk(n) x − . q Q n x − ϕ(q) n x ≪ (q,aX≤)=1 n a X(mod≤ q) (n,qX≤)=1 ≡ If the Lindelöf hypothesis is true for all Dirichlet L-functions, then the right-hand side 1 η can be replaced by x − . 38 SARY DRAPPEAU

In order to simplify the presentation, we put x if the generalized Lindelöf hypothesis is assumed, = E x1/k unconditionally.  To handle the small conductor case, we require the following. Lemma 7.2. For some δ > 0 and any non-principal character χ (mod q) with q x, of conductor r δ we have ≤ ≤E δ τk(n)χ(n) k x − . n x ≪ E X≤ Proof. Starting from the representation 1+1/(log x)+i s 1 ∞ k x ds τk(n)χ(n)= L(s, χ) (x N), n x 2πi Z1+1/(log x) i s 6∈ X≤ − ∞ one may truncate the contour at T = xδ/k, and shift it to the abscissa Re(s)=1 δ/k. The convexity bound L(1 δ/k + it, χ) qε(r( t +1))cδ/k+ε (for some c> 0)− yields the desired estimate if| =−x1/k. If the Lindelöf|≪ hypothesis| | L( 1 + it, χ) (q( t +1))ε E 2 ≪ | | is true, then one chooses T = xδ and shifts the contour to Re(s)=1 δ, where the bound L(1 δ + it, χ) (q( t + 1))ε holds by convexity. − − ≪ | | 7.1.1. Small conductors. Let 0 denote the quantity in the left-hand side of (7.1), and let R δ. The contributionS of those characters χ having conductors at most R is ≤E 1 χ(a) τk(n)χ(n). 1

δ 1 δ 2 x − x − R(log x) . E r R χ (mod r) q Q ϕ(q) ≪ E X≤ χ primitiveX Xr≤q | Letting R = δ/2, this is an acceptable error term. There remains to bound E 1 := τk(n)uR(na; q). S q Q n x (q,aX≤)=1 X≤

7.1.2. Dyadic decomposition. We dyadically decompose in 1 the sums over q and n in (7.1), yielding an upper bound S

2 (7.2) 1 (log x) sup τk(n)uR(na; q) . S ≪ Q′ x1/2+η Q′ 0 and assume throughout that δ is small with respect to η. When N x − , by the triangle inequality, our trivial bound (5.2) and Lemma 3.2, the sum≤ over q 1 η/2 1 η and n above is Ok(x − ), so we may add the restriction N > x − in the supremum 1/2+η 1/2+2η with an acceptable error. Then we relax the condition Q′ x into Q′ N . ≤ ≤ Renaming N into x, and expanding out τk(n), we obtain that it will suﬃce to prove (7.3) := u (n n a; q) x η S2 R 1 ··· k ≪ E− Q

2η 1/2+2η under the constraints a x and Q x . We decompose the sums over n1,...,nk dyadically to obtain an| | upper ≤ bound ≤

(7.4) := (log x)k sup u (n n a; q) . S2 ≪ S3 R 1 ··· k N1,...,Nk 1/2 Q

7.1.3. Splitting cases. Let the parameter 0 < δ1 < 1/100 be ﬁxed. We separate into two cases according to whether there is a subset 1,...,k such that J ⊂{ } δ1 1/3 δ1 Nj (x , x − ], j ∈ Y∈J or not. Suppose there is no such subset, and let

1/3 δ1 := j : 1 j k, N > x − . K { ≤ ≤ j } δ1 Necessarily card 3. Since Nj x for each j , and by assumption there is K ≤ ≤ 6∈δ K1 1/3 δ1 no subset 1,...,k r such that j Nj (x , x − ], it is necessarily the L⊂{ } K ∈L ∈ case that Q δ1 Nj x . j ≤ Y6∈K This implies card 1. Deﬁne K ≥ := (u ) CN : u 1 (n 1) . W { n ∈ | n| ≤ ≥ } Summarizing the above, we have

(7.5) xε( + + + ), S3 ≪k,ε A B3 B2 B1 where

= sup αmβnuR(nma; q) , A δ 1/3−δ x 1

3 = sup αmuR(n1n2n3ma; q) , 1/3−δ B N1,N2,N3>x 1 Q

2 = sup αmuR(n1n2ma; q) , B 1/3−δ1 N1,N2>x Qx 1 (q,a)=1 M/8

1 = sup αmuR(nma; q) . B N>x1−δ1 Q

We will focus on and 3, since the treatment of 1 and 2 is analogous to 3 and actually simpler. A B B B B 40 SARY DRAPPEAU

7.1.4. Separation of variables. Fix another small parameter δ2 > 0. We smoothen the cutoff using a smooth function φ : R [0, 1] with φ(ξ)=1 for ξ [1, 2], φ(ξ)=0 δ2 δ2 → (j) ∈jδ2 for ξ [1 − , 2+ − ], whose derivatives satisfy φ j . The cost of 6∈ −E E k k∞ ≪ E replacing in and 3 the sharp cutoff condition x < nm 2x (resp. x < n1n2n3m A B ≤δ2/2 ≤ 2x) by φ(nm/x) (resp. φ(n1n2n3m/x)) is at most O(x − ), by trivially bounding the contribution of the transition ranges using Lemma 3.2.E ˘ s 1 Integration by parts shows that the Mellin transform φ(s)= 0∞ φ(ξ)ξ − dξ satisfies

5δ2 R φ˘(it) E (t R). ≪ 1+ t 5 ∈ | | 1 it We use the inversion formula φ(ξ) = (2π)− R φ˘(it)ξ− dt at ξ = nm/x (resp. ξ = mn n n /x) in the case of (resp. ), to obtain the upper bounds 1 2 3 A B3 R

δ2/2 5δ2 (7.6) k x − + sup αmβnuR(mna; q) , A≪ E E δ 1/3−δ x 1

δ2/2 5δ2 1 − 3 k x + sup 3 1/3−η B ≪ E E N1,N2,N3>x , 1+ t × (αm) , t R | | (7.7) ∈W ∈ it αm(n1n2n3) uR(n1n2n3ma; q) . × Q

7.1.5. The case of . Let (αm), (βn) and N be given as in the supremum in (7.6). We wish to bound A

(7.8) := α β u (mna). Sa m n R Q

By dyadic decomposition, enlarging our bound by a factor of k2, we may assume k k+1 the conditions are N < n 2N and M < m 2M for M N [x2− , x2 ]. 1 ≤ 1 1 ≤ 1 1 1 ∈ Theorem 5.1 with η min δ1, 1/30 gives the existence of δ3 > 0 depending on δ1 ← { k } δ3 k δ3 such that (7.8) is majorized by O(2 x − ), on the condition that a 2− x and Q k 1/2+δ3 E | | ≤ ≤ 2− x , which are satisﬁed assuming η < δ3/4 and taking x large enough in terms of k. 1/3 If on the contrary Q x , we appeal to Lemma 5.2 with η δ1/k (or η δ1 if the Lindelöf hypothesis≤ is assumed). We again obtain for (7.8) a← bound ←

k δ3 2 x − Sa ≪j E for some δ3 (depending on δ1). Summarizing, we have obtained in any case

δ2/2 5δ2 δ3 (7.9) x − + x − A≪k E E for δ3 > 0. Choosing δ2 appropriately, it is an acceptable error term once we can prove that δ1 > 0 can be chosen independently of k. ERROR TERM IN THE DISPERSION METHOD 41

1/3 δ1 7.1.6. The case of 3. Let (αm), N1, N2, N3 > x − and t R be as in supremum in (7.7). The quantityB we wish to bound is at most ∈

1 itu b := 3 (n1n2n3) R(n1n2n3ma; q) S 1+ t M/8 m 2M Q

3δ1 it it 2Nj it 1 where N N N M = x and M < x . Writing n = (2N ) it z − dz, the above 1 2 3 j j − nj is bounded by R

(7.10) b ε sup uR(n1n2n3ma) S ≪ ′ ′ ′ N1,N2,N3 M/8 m M Q

Fix N1′ , N2′ , N3′ as in the supremum. Using (5.3) and the triangle inequality,

′ + ′′, Sb ≤ Sb Sb where u (7.11) b′ = 1(n1n2n3ma) , S M/8 m M QWeil conjectures [Del74]). For 1/2+δ4 some small, absolute δ4, on the condition that Q x (requiring η < δ4/2), the quantity (7.11) is bounded by ≤

1 δ4 1 δ4+3δ1 (7.13) ′ Mx − x − . Sb ≪ ≤ ε 1/2 Consider then b′′. By Lemma 3.4, each sum over n is bounded by Oε(x R ), and so we obtain a boundS ε 5/2 ′′ x R M Sb ≪ε which is absorbed in the term (7.13). Inserting in (7.7), we have obtained for 3 a bound B

δ2 5δ2 1 δ4+3δ1 (7.14) x − + x − . B3 ≪ E E The terms 2 and 1 are shown in the same way to satisfy the same bound with δ4 > 0 absoluteB and smallB enough. Choosing our parameters adequately, we can choose absolute constants δ1, δ2, δ3 in such a way that both bounds (7.14) and (7.9) are true η and O(x − ). Inserting back into (7.5) and (7.4), we obtain the claimed bound (7.3). E 7.2. Proof of Theorems 1.5 and 1.6. As a last step, we deduce from Theorem 7.1 the estimate

1 η η (7.15) τk(n) τk(n) k x − (0 < a x ) 2 − ϕ(q) 2 ≪ E | | ≤ qX√x nXq nXq (q,a≤)=1 n a (mod≤ q) (n,q≤)=1 ≡ 42 SARY DRAPPEAU where as before = x if the generalized Lindelöf is true and = x1/k otherwise. Let ∆ (0, 1/10)Ebe ﬁxed and decompose the sums over q and nEinto intervals ((1 + 1 ∈ 1 ∆)− Q, Q] and ((1 + ∆)− N, N]. Calling ′ the left-hand side of (7.15), we have S1

′ τ (n)u (na; q) , S1 ≪ k 1 j0,j1 0 (1+∆)−1Q

k 2 2 u 1′ ∆x(log x) + (log x) ∆− sup τk(n) 1(na; q) . S ≪ Q √x (1+∆)−1Q 0 be the real number given by Theo- rem 7.1. Lemma 3.2 gives the bound

ε τk(n)u1(na; q) ε x N −1 −1 ≪ (1+∆)XQ

1/2+ε k(x) =2 τk(n)+ Oε(x ) T 2 qX√x q

1 δ/k The main terms are computed in [FT85, Théorème 2], with an error term O(x − ) (unconditionally). If one assumes the generalized Lindelöf hypothesis, then the proof is adapted in the following way. Under the hypotheses and in the notations of [FT85, ν δ k Lemma 6], there holds θ(p ) Cp− k/2 ([FT85, ﬁrst display page 52]). There- | | ≤ ⌊ ⌋ fore the series Fk(s) in [FT85, Lemma 7] is bounded in terms of k only in the half- place Re(s) 1 δ/2. In the proof of [FT85, Lemma 7], one chooses T = xδ/2 and shift the contour≥ − to Re(s)=1 δ/2, where the Lindelöf hypothesis implies ζ(s) tε by convexity, to produce the conclusion− ≪

1 δ/2+ε Ψ(n)τk(n)= xQk 1(log x)+ Oε,k(x − ). − n x X≤ ERROR TERM IN THE DISPERSION METHOD 43

The rest of the argument in Corollaries 1-2 of Lemma 7, and Corollary of Lemma 8 of [FT85] are transposed verbatim to yield

1 1 c 2 τk(n)= xPk(log x)+ Ok(x − ) ϕ(q) 2 qX√x q 0, as claimed. 7.3. Remark on the uniformity in a. If we were to replace the shift τ(n + 1) by τ(n + a), 0 < a xδ, then the deduction of an asymptotic formula analogous to (1.4) from Theorem| | ≤ 7.1 goes along similar lines. We brieﬂy indicate how one reduces to our previous setting. From Dirichlet’s hyperbola method, the problem reduces to the evaluation of k,a(x)=2 τk(n). S 2 qX√x q Xn x ≤ n a≤(mod≤ q) ≡− Extracting the largest factor d a∞ from n, we rewrite this as 1| k,a(x)=2 τk(d1) τk(n). S ∞ 2 d1 a q √x q /d n x/d X| X 1 X 1 ≤ (n,a≤ )=1≤ nd1 a (mod q) ≡− Writing d := (q,d ), the congruence condition is equivalent to d a and 2 1 2| n (a/d )(d /d ) (mod q/d ). ≡ − 2 1 2 2 We therefore have

k,a(x)=2 τk(d1) τk(n). S ∞ 2 d1 a d2 (d1,a) q √x/d2 q /d1 n x/d1 X| |X ≤X ≤X≤ (q,d1/d2)=(q,a/d2)=1 (n,a)=1 n (a/d2)(d1/d2) (mod q) ≡− δ Summing for each dj individually, the contribution of d1 > x is bounded trivially using Lemma 3.2. When d xδ, the sum over n and q is handled by an adequate 1 ≤ generalization of Theorem 7.1, involving a congruence of the type n b1b2 (mod q), ≡ δ as well as an additional coprimality condition (n, b3)=1, for integers bj x . Our arguments readily adapt to account for both these modiﬁcations. Note| however| ≤ that it is now important that the method is able to handle values of the modulus q up to x1/2+δ, with δ independent of k (cf. the statement of Theorem 7.1).

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