Estimates for character sums and Dirichlet L -functions to smooth moduli
EEstimates for character sums and Dirichlet L -functionsto smooth moduli A.J. IrvingCentre de recherches math´ematiques, Universit´e de Montr´eal
Abstract
We use the q -analogue of van der Corput’s method to estimate short charactersums to smooth moduli. If χ is a primitive Dirichlet character modulo a squarefree, q δ -smooth integer q we show that L ( 12 , χ ) (cid:28) (cid:15) q + O ( δ )+ (cid:15) . Suppose χ is a nontrivial Dirichlet character to modulus q . An important question in analyticnumber theory is the estimation of the character sum S = (cid:88) M
3. In the former situationit gives a power-saving for S provided that N ≥ q + δ , for some δ >
0. Taking r = 2 and N (cid:28) √ q the Burgess estimate becomes S (cid:28) (cid:15) q + (cid:15) . Burgess deduced from this that L ( 12 , χ ) (cid:28) (cid:15) q + (cid:15) . This improves on the trivial “convexity” bound of q .Burgess’s result is currently the sharpest known for arbitrary moduli, in particular it hasnot been improved for prime q . However, by using methods based on Weyl differencing,various improvements can be given if we make assumptions on the prime factorisation of q .In particular better results are known if q is either powerful or smooth.We say that q is powerful if (cid:89) p | q p is small relative to q . In that case the sum may be estimated by a result of Iwaniec [15],which extends previous work on prime-power moduli of Postnikov [18] and Gallagher [8].These works contain various applications to L ( s, χ ) when (cid:60) s is close to 1. In a recent paper[17], Mili´cevi´c developed a quite general form of p -adic van der Corput method. When q isa prime power this can be used to give estimates for S which are completely analogous tothose arising from the classical theory of exponent pairs. If q is a prime power, q = p n , thenMili´cevi´c obtained the estimate L ( 12 , χ ) (cid:28) (cid:15) p r q θ + (cid:15) , for some fixed r and θ ≈ . q , for which the first result was given byHeath-Brown in [12]. He showed that if q = q q is squarefree then S (cid:28) (cid:15) q (cid:15) (cid:16) √ N q + √ N q (cid:17) . (2)2e say that q is q δ -smooth if all the prime factors of q are at most q δ . In that case we canfind a factorisation q = q q with q ∈ [ q , q + δ ] and (2) becomes S (cid:28) (cid:15) √ N q + O ( δ )+ (cid:15) . (3)If N (cid:28) √ q then this gives S (cid:28) (cid:15) q + O ( δ )+ (cid:15) which is better than the Burgess bound for sufficiently small δ . Heath-Brown deduced fromthis that L ( 12 , χ ) (cid:28) (cid:15) q + O ( δ )+ (cid:15) . This is analogous to Weyl’s estimate ζ ( 12 + it ) (cid:28) (cid:15) t + (cid:15) . The key idea in proving the estimate (2) is to apply van der Corput differencing (calledthe A -process) to reduce the modulus of the sum from q to q . This latter sum may then becompleted, which is an analogue of the van der Corput B -process, to give the result. Grahamand Ringrose [11] generalised this procedure, giving a q -analogue of the van der Corput A k B estimate. A consequence of their result is that for any η > δ > q is q δ smooth and N ≥ q η then one can give a power-saving for S . This has applications tothe distribution of zeros near (cid:60) s = 1 of L ( s, χ ), but not to L ( , χ ) since the optimal choiceof k is then 1.By combining the methods for powerful and smooth q one can obtain results for a largerset of moduli. This was recently studied by Chang [5], in which the object is to obtain anontrivial bound for N as small as possible. On the other hand Goldmakher [9] studied longsums, that is N almost as large as q . He demonstrated that for certain q a small improvementon the P´olya-Vinogradov inequality is possible.The aim of this paper is to prove an estimate for S which, when q is sufficiently smooth,improves on (3) when N ≈ √ q . For such q we will deduce an estimate for L ( , χ ) whichbreaks the Weyl barrier of q . Theorem 1.1.
Suppose that q is squarefree and q δ smooth for some δ > . Then, for anyprimitive Dirichlet character χ to modulus q , any N ≤ q and any (cid:15) > we have S = (cid:88) M ≤ n ≤ M + N χ ( n ) (cid:28) (cid:15) N q + O ( δ )+ (cid:15) . Theorem 1.2.
For q and χ as in Theorem 1.1 and any (cid:15) > we have L ( 12 , χ ) (cid:28) (cid:15) q + O ( δ )+ (cid:15) . q -analogue of the van der Corput ABA B process. This is unsurprising since the original van der Corput ABA B estimate can be usedto obtain the bound ζ ( 12 + it ) (cid:28) (cid:15) t + (cid:15) , see Graham and Kolesnik [10, Theorem 4.1] for a proof. The proof of Theorem 1.1 has manysimilarities with our work [14], in which a q -analogue of BA k B was used to estimate shortKloosterman sums. However, the addition of an initial A -process leads to some technicalchanges. As in [14] a crucial role is played by an estimate for a complete multidimensionalexponential sum. This takes a different form to that in [14] due to the replacement ofKloosterman fractions by Dirichlet characters as well as the extra differencing step. The keyestimate is proved in an appendix by Fouvry, Kowalski and Michel, using tools developed intheir work [7].The exponents in Theorem 1.1, and similar results, can be predicted by means of thetheory of exponent pairs. This is described, in the context of the classical van der Corputmethod, by Graham and Kolesnik in [10, Chapter 3]. Suppose that ( k, l ) is an exponentpair derived from the trivial pair (0 ,
1) by applications of the A and B processes. We thenexpect that by applying the q -analogue of the same sequence of processes we can establishthe bound (cid:88) M ≤ n 1) = ( , ). Similarly, Heath-Brown’s estimate (3) corresponds to the pair AB (0 , 1) = ( , ).Our exponent ≈ . . q ,whereas a positive proportion of q will be sufficiently smooth and therefore satisfy the hy-potheses of Theorem 1.2. In order to obtain the same exponent as Mili´cevi´c we would need toiterate the A and B processes an arbitrary number of times, thereby obtaining a sufficientlygeneral form of (4). This would be technically challenging, but a more significant issue wouldbe to give bounds for the progressively more complex complete sums which would arise.Throughout this work we adopt the standard convention that (cid:15) denotes a small positivequantity whose value may differ at each occurrence. If f is a function on Z /q Z then we will,without comment, extend f to a function on Z (or on Z /q (cid:48) Z for any multiple q (cid:48) of q ). Acknowledgements This work was completed whilst I was a CRM-ISM postdoctoral fellow at the Universit´e deMontr´eal. I am very grateful to professors Fouvry, Kowalski and Michel for their appendixproving the crucial complete sum estimate. 4 The A -Process In this section we will describe the q -analogue of the van der Corput A k -process for quitegeneral functions f . We begin with a single iteration. When f is a Dirichlet character thefollowing lemma was first proved by Heath-Brown in [12], it has since been modified for anumber of different f . A result with a similar level of generality to ours was given by Blomerand Mili´cevi´c in [1, Lemma 12]. Lemma 2.1. Suppose q = q q . For i ∈ { , } let f i : Z /q i Z → C with f i ( n ) (cid:28) for all n .Let f ( n ) = f ( n ) f ( n ) be a function on Z /q Z and let I be an interval of length at most N .We then have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) n ∈ I f ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) q N + (cid:88) < | h |≤ N/q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) n ∈ I ( h ) f ( n ) f ( n + q h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where I ( h ) are subintervals of I .Proof. If N < q then q N > N so the result follows from the trivial bound (cid:88) n ∈ I f ( n ) (cid:28) N. We therefore assume, for the remainder of the proof, that N ≥ q . We let a n = (cid:40) f ( n ) n ∈ I S = (cid:88) n a n . Since N ≥ q we have H = [ N/q ] ≥ f has period q , we obtain HS = H (cid:88) h =1 (cid:88) n a n + q h = (cid:88) n f ( n ) (cid:88) h =1 n + q h ∈ I f ( n + q h ) . The outer summation is supported on an interval of length O ( N ) so we may use Cauchy’sinequality and the bound f ( n ) (cid:28) H S ≤ N (cid:88) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) h =1 n + q h ∈ I f ( n + q h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = N H (cid:88) h ,h =1 (cid:88) nn + q h i ∈ I f ( n + q h ) f ( n + q h ) . H S ≤ N H (cid:88) h ,h =1 (cid:88) n ∈ In + q ( h − h ) ∈ I f ( n ) f ( n + q ( h − h )) (cid:28) N H (cid:88) | h |≤ H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) n ∈ I ( h ) f ( n ) f ( n + q h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where I ( h ) is a subinterval of I . We apply the bound f ( n ) (cid:28) h = 0and conclude that S (cid:28) N H − N + (cid:88) < | h |≤ H (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) n ∈ I ( h ) f ( n ) f ( n + q h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The result follows since N H − (cid:28) q .For any complex-valued function f we introduce the notation f ( n ; h , . . . , h k ) = (cid:89) I ⊆{ ,...,k } f (cid:32) n + (cid:88) i ∈ I h i (cid:33) σ ( I ) , where σ ( I ) denotes that the complex conjugate is taken when | I | is odd. The term I = ∅ isto be included in the product. The result of the last lemma may therefore be written (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) n ∈ I f ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) q N + (cid:88) < | h |≤ N/q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) n ∈ I ( h ) f ( n ; q h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Iterating this k times we derive the following. Lemma 2.2. Suppose that k ∈ N and q = q q . . . q k . For ≤ i ≤ k let f i : Z /q i Z → C befunctions with f i ( n ) (cid:28) and let f ( n ) = (cid:81) ki =0 f i ( n ) . If I is an interval of length at most N then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) n ∈ I f ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:28) k k (cid:88) j =1 N k − k − j q k − j k − j +1 + N k − k − ( q/q ) (cid:88) < | h |≤ N/q . . . (cid:88) < | h k |≤ N/q k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) n ∈ I ( h ,...,h k ) f ( n ; q h , . . . , q k h k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . In the last line I ( h , . . . , h k ) denotes a subinterval of I and f ( n ; q h , . . . , q k h k ) was definedabove. roof. The result can be proved by induction using Lemma 2.1. The details, for specificchoices of f , have previously been given in Heath-Brown [13] and our paper [14]. The proofis unchanged in the current, more general, setting so we do not include it. B -Process The q -analogue of the van der Corput B -process is more commonly known simply as com-pletion of a sum. It has been known, in some form, at least since the work of P´olya andVinogradov. In order to state it we define the Fourier transform ˆ f : Z /q Z → C of a function f : Z /q Z → C by ˆ f ( x ) = 1 √ q (cid:88) n (mod q ) f ( n ) e q ( nx ) , where e q ( y ) = e πiyq . Lemma 3.1. Suppose we are given a function f : Z /q Z → C and an interval I ⊆ [ M, M + N ) with N ≤ q . We assume, without loss of generality, that M ∈ Z . We then have (cid:88) n ∈ I f ( n ) (cid:28) N ˆ f (0) √ q + N log q √ q max J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) x ∈ J ˆ f ( x ) e q ( − M x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where the maximum is taken over all intervals J of length at most q/N which do not contain .Proof. The Plancherel identity gives S := (cid:88) n ∈ I f ( n ) = (cid:88) x (mod q ) ˆ f ( x ) ˆ I ( x ) , where ˆ I is the Fourier transform of the indicator function of I . We can writeˆ I ( x ) = 1 √ q (cid:88) n ∈ I e q ( nx ) = e q ( M x ) 1 √ q (cid:88) n ≤ NM + n ∈ I e q ( nx ) = e q ( M x ) g ( x ) , say. Since g is given by a geometric series we have the well-known estimate g ( x ) (cid:28) √ q min( N, (cid:107) x/q (cid:107) ) , in which (cid:107) . (cid:107) denotes the distance from a real number to the nearest integer. In simpleapplications of completion this is used in conjunction with an estimate for the completesums ˆ f ( x ). In this work however, as in [14], we wish to be able to detect cancellation7etween the ˆ f ( x ) by means of applications of the A -process. We therefore split the sum over x into short intervals, on which the factor g ( x ) may be removed by partial summation.We identify the summation range x (mod q ) with the integers in ( − q , q ] and extend g ( x )to a smooth function of x on that interval. We therefore write S (cid:28) N ˆ f (0) √ q + (cid:88) − q/ 0. We may therefore dividethe sum into O ( N ) intervals of this form and conclude that S (cid:28) N ˆ f (0) √ q + N log q √ q max J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) x ∈ J ˆ f ( x ) e q ( − M x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where the maximum is as described in the statement of the lemma. The term log q arisesfrom the estimate (cid:88) r (cid:28) N r (cid:28) log N + 1 (cid:28) log q. We also give a version of the last lemma which is less precise but which allows the lengthof the sum to exceed the modulus q . This is essentially the usual completion of sums, usedby P´olya and Vinogradov etc, so we do not include a proof. An almost identical result isgiven by Iwaniec and Kowalski in [16, Lemma 12.1]. Lemma 3.2. Suppose we are given a function f : Z /q Z → C and an interval I ⊆ [ M, M + N ) . We then have (cid:88) n ∈ I f ( n ) (cid:28) N ˆ f (0) √ q + 1 √ q (cid:88) x (cid:54)≡ q ) | ˆ f ( x ) |(cid:107) x/q (cid:107) . Complete Sums In the course of our proof of Theorem 1.1 we will need to deal with two different completesums to a squarefree modulus. In both cases the Chinese Remainder Theorem is used towrite the sum as a product of complete sums to prime moduli. The latter are then boundedby an appeal to a suitable form of the Riemann hypothesis over finite fields. The first sumis 1-dimensional so the necessary result follows from the work of Weil [19]. The second sumis much more complex. The necessary result can be derived from the work of Deligne [6],the details are given by Fouvry, Kowalski and Michel in the appendix. -Dimensional Sum Suppose χ (mod q ) is a primitive Dirichlet character. For integers h and x we consider thesum W χ,h ( x ) = (cid:88) n (mod q ) χ ( n ) χ ( n + h ) e q ( nx ) . A more general form of this sum was discussed by Graham and Ringrose in [11, Section 4].We begin by observing that if ( a, q ) = 1 then W χ,h ( ax ) = (cid:88) n (mod q ) χ ( n ) χ ( n + h ) e q ( anx )= (cid:88) n (mod q ) χ ( an ) χ ( an + h ) e q ( nx )= (cid:88) n (mod q ) χ ( n ) χ ( n + ah ) e q ( nx )= W χ,ah ( x ) . Combining this with a special case of [11, Lemma 4.1] we can deduce the following multi-plicative property. Lemma 4.1. Suppose q = uv with ( u, v ) = 1 and that χ is a primitive Dirichlet charactermodulo q . Then there exist primitive characters χ u (mod u ) and χ v (mod v ) such that W χ,h ( x ) = W χ u ,vh ( x ) W χ v ,uh ( x ) where uu ≡ v ) and vv ≡ u ) . Using the results of Weil [19], Graham and Ringrose [11, Lemmas 4.2 and 4.3] obtainedthe following estimate for the sum to a prime modulus. Our bound is slightly weaker thantheirs as we do not include their better estimate when p | h but p (cid:45) x .9 emma 4.2. If p is prime and χ is a primitive Dirichlet character modulo p then for anyintegers h and x we have W χ,h ( x ) (cid:28) (cid:40) p h ≡ x ≡ p ) √ p otherwise. Observe that the bound in the lemma can be expressed more compactly as p ( h, x, p ) .We may therefore combine our results to obtain a bound for squarefree moduli. Lemma 4.3. Suppose q is squarefree and χ (mod q ) is a primitive character. Then, for anyintegers h, x and any (cid:15) > we have W χ,h ( x ) (cid:28) (cid:15) q + (cid:15) ( h, x, q ) . We now turn our attention to a complete sum constructed from products of the sums W .Recall that for a function f ( n ) and integers h , . . . , h k we defined f ( n ; h , . . . , h k ) = (cid:89) I ⊆{ ,...,k } f (cid:32) n + (cid:88) i ∈ I h i (cid:33) σ ( I ) . Using this notation we let K χ,h ( h , . . . , h k , y ) = (cid:88) x (mod q ) e q ( xy ) W χ,h ( x ; h , . . . , h k ) . We begin by establishing a multiplicative property. Lemma 4.4. If χ is a primitive character mod q = uv with ( u, v ) = 1 then K χ,h ( h , . . . , h k , y ) = K χ u ,vh ( h , . . . , h k , vy ) K χ v ,uh ( h , . . . , h k , uy ) . Proof. By Lemma 4.1 we obtain W χ,h ( x ; h , . . . , h k ) = W χ u ,vh ( x ; h , . . . , h k ) W χ v ,uh ( x ; h , . . . , h k ) . The result then follows from the Chinese Remainder Theorem.It remains to estimate the sums K when the modulus is prime. Lemma 4.5. Let χ be a primitive Dirichlet character modulo a prime p , suppose h (cid:54)≡ p ) and let h , . . . , h k , y be integers. . If y (cid:54)≡ p ) or if k (cid:89) i =1 h i (cid:54)≡ p ) then K χ,h ( h , . . . , h k , y ) (cid:28) k p k +12 . 2. Otherwise we have the trivial estimate K χ,h ( h , . . . , h k , y ) (cid:28) k p k +22 . Proof. By increasing the value of the implied constant in the result we may assume that p > 5. Let S ( y ) be the sum defined in the appendix with n = n = 2 k − and with shifts t i , s j given by (cid:80) i ∈ I h i ( t i corresponding to | I | even and s j to | I | odd). Wethen have K χ,h ( h , . . . , h k , y ) = p k − S ( y )so Theorem 1 from the appendix may be applied. Since n = n we obtain S ( y ) (cid:28) k √ p provided that either y (cid:54)≡ p ) or that one of the shifts (cid:80) i ∈ I h i occurs an odd numberof times modulo p . However, we showed in [14, Lemma 4.5] that if the latter condition failsthen h i ≡ p ) for at least one index i and thus k (cid:89) i =1 h i ≡ p ) . This completes the proof of the first estimate and the second follows directly from Lemma4.2.Combining the last two lemmas we deduce a bound for K when q is squarefree. Lemma 4.6. Suppose χ is a primitive Dirichlet character modulo the squarefree integer q , ( h, q ) = 1 and let h , . . . , h k , y be integers. Then, for any (cid:15) > , we have K χ,h ( h , . . . , h k , y ) (cid:28) (cid:15),k q k +12 + (cid:15) (cid:32) q, y, k (cid:89) i =1 h i (cid:33) . Theorem 1.1: First Steps Before beginning the proof of Theorem 1.1 we give the following well-known lemma. Lemma 5.1. Suppose q is an integer and H ≥ . For any (cid:15) > we have (cid:88) We have (cid:88) Suppose that q = q q . . . q k is squarefree, χ is a primitive Dirichlet charactermodulo q and ( h, q ) = 1 . Let I be an interval of length at most N ≤ q and consider the sum T = (cid:88) n ∈ I χ ( n ) χ ( n + h ) . For any (cid:15) > we have T (cid:28) (cid:15),k q (cid:15) (cid:32) k (cid:88) j =1 N − j q / − − j q − j k − j +1 + N − k q / − − k q / k +1 (cid:33) . The last lemma should be compared to our estimate [14, Theorem 1.3], in which a similarbound was obtained for short Kloosterman sums. Both results use a q -analogue of the vander Corput BA k B estimate and the proofs are essentially the same apart from the use ofdifferent estimates for complete exponential sums. In [14] the result contains an extra term q / q − / k +1 . In the course of the proof of Lemma 5.2 we will show that this term is notneeded; the same argument could be used to remove it from [14, Theorem 1.3]. In theremainder of this section we show that Lemma 5.2 implies Theorem 1.1, we will prove theformer in the next section.The next result specialises Lemma 5.2 to k = 3 and smooth moduli q . Lemma 5.3. Suppose q is squarefree and q δ -smooth for some δ > . Let h , χ , I , N and T be as in Lemma 5.2. Assuming that δ is sufficiently small we have T (cid:28) (cid:15) N q + O ( δ )+ (cid:15) . Proof. If N ≤ q then N q ≥ N so the result follows from the trivial bound | T | ≤ N .For the remainder of the proof we assume that N ≥ q . We apply Lemma 5.2 with k = 3and a factorisation q = q q q q for which q j ≈ Q j , where Q = q − N ,Q = q − N , = q N − and Q = q N − . Observe that Q Q Q Q = q . Furthermore, by our assumptions that q ≤ N ≤ q , weknow that all the Q j exceed some fixed power of q . It follows that if q is q δ -smooth, with δ sufficiently small, then we can find a factorisation with q j ∈ [ Q j q − δ , Q j ] for j = 1 , , q ∈ [ Q , Q q δ ] . Note that this choice is not quite optimal if we wish to get the best dependence on δ . Lemma5.2 now gives the required result.Observe that the estimate of Lemma 5.3 may be written T (cid:28) (cid:15) q O ( δ )+ (cid:15) ( q/N ) N and that ( , ) = BA B (0 , T may also be estimated by a direct applicationof Lemmas 3.2 and 4.3. This gives the following result, which can handle the case N > q . Lemma 5.4. Suppose q is squarefree, ( h, q ) = 1 and χ (mod q ) is a primitive Dirichletcharacter modulo q . Let I be an interval of length at most N and T as in the previouslemmas. We then have T (cid:28) (cid:15) (cid:18) Nq + 1 (cid:19) q + (cid:15) . We now generalise to the case ( h, q ) > 1. We do not give an optimal treatment, preferringto give a simple estimate which is sufficiently sharp to prove Theorem 1.1. Lemma 5.5. Suppose that χ is a primitive Dirichlet character modulo the squarefree, q δ -smooth integer q and that I is an interval of length at most N ≤ q . For any integer h andany (cid:15) > we have T (cid:28) (cid:15) (cid:112) ( h, q ) N q + O ( δ )+ (cid:15) . Proof. We write ( h, q ) = d and q = de (recall ( d, e ) = 1 since q is squarefree). Factorising14 = χ d χ e for primitive characters χ d (mod d ) and χ e (mod e ) we obtain T = (cid:88) n ∈ I χ ( n ) χ ( n + h )= (cid:88) n ∈ I χ d ( n ) χ e ( n ) χ d ( n + h ) χ e ( n + h )= (cid:88) n ∈ I | χ d ( n ) | χ e ( n ) χ e ( n + h )= (cid:88) n ∈ I ( n,d )=1 χ e ( n ) χ e ( n + h )= (cid:88) d (cid:48) | d µ ( d (cid:48) ) (cid:88) n ∈ Id (cid:48) | n χ e ( n ) χ e ( n + h ) ≤ (cid:88) d (cid:48) | d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) n ∈ I/d (cid:48) χ e ( n ) χ e ( n + d (cid:48) h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . In the final line I/d (cid:48) = { x : d (cid:48) x ∈ I } and d (cid:48) d (cid:48) ≡ e ) . We have two bounds for the final sum, which has length at most N/d (cid:48) ≤ N and modulus e .Firstly, Lemma 5.4 gives (cid:88) n ∈ I/d (cid:48) χ e ( n ) χ e ( n + d (cid:48) h ) (cid:28) (cid:15) (cid:18) Ne + 1 (cid:19) e + (cid:15) . Secondly, we can use Lemma 5.3. Since e divides q we know that it is q δ smooth. Therefore,if e ≥ q O (1) then it is e O ( δ ) smooth. Therefore, provided that N ≤ e , we obtain a bound N e + O ( δ )+ (cid:15) ≤ N q + O ( δ )+ (cid:15) . Combining these two estimates we obtain (cid:88) d (cid:48) | d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) n ∈ I/d (cid:48) χ e ( n ) χ e ( n + d (cid:48) h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:28) (cid:15) N q + O ( δ )+ (cid:15) + N √ e and thus T (cid:28) (cid:15) N q + O ( δ )+ (cid:15) + N (cid:112) ( h, q ) √ q ≤ (cid:112) ( h, q ) (cid:18) N q + O ( δ )+ (cid:15) + N √ q (cid:19) . The first term is largest since we are assuming that N ≤ q and thus the result follows.15e now return to the sum S . Recall that we have shown that S (cid:28) q N + (cid:88) < | h |≤ N/q | S ( q h ) | where S ( q h ) = (cid:88) n ∈ I ( h ) χ ( n ) χ ( n + q h ) . The quantity q is q δ smooth and therefore q O ( δ )0 -smooth provided that q ≥ q O (1) . Assuming N ≤ q we use Lemma 5.5 to obtain S (cid:28) (cid:15) q O ( δ )+ (cid:15) q N + N q (cid:88) < | h |≤ N/q (cid:112) ( h, q ) (cid:28) (cid:15) q O ( δ )+ (cid:15) (cid:16) N q + N q (cid:17) , the final estimate following from Lemma 5.1.We now let Q = N − q and Q = N q . Observe that Q Q = q and that, for N ≤ q , Q , Q ≥ q O (1) . Therefore, if δ is sufficientlysmall, we may factorise q = q q with q ∈ [ Q , Q q δ ]and q ∈ [ Q q − δ , Q ] . By our assumption (5) on the size of N we know that N ≤ q and therefore the abovearguments give S (cid:28) (cid:15) N q + O ( δ )+ (cid:15) . This completes the proof of Theorem 1.1. We begin by proving the result in the case qN ≤ q . Suppose that I ⊆ [ M, M + N ]. We applythe B -process, Lemma 3.1, with f ( n ) = χ ( n ) χ ( n + h ), to obtain T (cid:28) N ˆ f (0) √ q + N log q √ q max J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) x ∈ J ˆ f ( x ) e q ( − M x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , J of length at most q/N which do not contain0. The Fourier transform is given byˆ f ( x ) = 1 √ q (cid:88) n (mod q ) χ ( n ) χ ( n + h ) e q ( nx ) = 1 √ q W χ,h ( x ) . We know that ( h, q ) = 1 and therefore Lemma 4.3 givesˆ f ( x ) (cid:28) (cid:15) q (cid:15) . We may thus deduce that N ˆ f (0) √ q (cid:28) (cid:15) N q − + (cid:15) which is certainly sufficiently small. Lemma 5.2 will therefore follow if we can show that (cid:88) x ∈ J √ q W χ,h ( x ) e q ( − M x ) (cid:28) (cid:15),k q + (cid:15) N (cid:32) k (cid:88) j =1 N − j q / − − j q − j k − j +1 + N − k q / − − k q / k +1 (cid:33) for all intervals J of length at most q/N . On writing K = q/N we see that it is sufficient toestablish (cid:88) x ∈ J √ q W χ,h ( x ) e q ( − M x ) (cid:28) (cid:15),k q (cid:15) (cid:32) k (cid:88) j =1 K − − j q − j k − j +1 + K − − k q / k +1 (cid:33) . This final estimate should be compared with existing A k B results, for example that givenby Heath-Brown in [13, Theorem 2]. In order to prove it we begin by applying Lemma 2.2with f ( x ) = 1 √ q W χ,h ( x ) e q ( − M x ) . By Lemma 4.1 and the Chinese Remainder Theorem we may factorise this as f ( x ) = k (cid:89) i =0 f i ( x )where f i ( x ) = 1 √ q i W χ i ,q/q i h ( x ) e q i ( − M q/q i x ) , χ i (mod q i ). We compute f ( x ; h , . . . , h k ) = (cid:89) I ⊆{ ,...,k } f (cid:32) x + (cid:88) i ∈ I h i (cid:33) σ ( I ) = q − k − W χ ,q/q h ( x ; h , . . . , h k ) (cid:89) I ⊆{ ,...,k } e q ( − M q/q ( x + (cid:88) i ∈ I h i ) σ ( I ) . Since I runs over the same number of even and odd subsets we have (cid:89) I ⊆{ ,...,k } e q ( − M x ) σ ( I ) = 1 . It follows that (cid:89) I ⊆{ ,...,k } e q ( − M q/q ( x + (cid:88) i ∈ I h i )) σ ( I ) depends only on the h i , not on x , and therefore it can be taken outside the sum in Lemma2.2. We obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) x ∈ J √ q W χ,h ( x ) e q ( − M x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:28) (cid:15),k q (cid:15) (cid:32) k (cid:88) j =1 K k − k − j q k − j k − j +1 + K k − k − ( q/q ) (cid:88) < | h |≤ K/q . . . (cid:88) < | h k |≤ K/q k q − k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) x ∈ J ( h ,...,h k ) W χ ,q/q h ( x ; q h , . . . , q k h k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We apply Lemma 3.2 to the final sum, followed by Lemma 4.6, to obtain (cid:88) x ∈ J ( h ,...,h k ) W χ ,q/q h ( x ; q h , . . . , q k h k ) (cid:28) k,(cid:15) q k − + (cid:15) K √ q (cid:32) q , k (cid:89) i =1 q i h i (cid:33) + 1 √ q (cid:88) x (cid:54)≡ q ) (cid:107) x/q (cid:107) (cid:32) q , x, k (cid:89) i =1 q i h i (cid:33) (cid:28) k,(cid:15) q k − + (cid:15) K √ q (cid:32) q , k (cid:89) i =1 h i (cid:33) + √ q (cid:88) 20e conclude that T (cid:28) (cid:15),k q (cid:15) (cid:32) k − l (cid:88) j =1 N − j q / − − j q − j k − j +1 + N l − k − q / − l − k − q l − k − l (cid:33) . = q (cid:15) k − l +1 (cid:88) j =1 N − j q / − − j q − j k − j +1 This is sharper than the required estimate and therefore completes the proof of Lemma 5.2. We modify the proof given by Iwaniec and Kowalski in [16, Theorem 12.9], replacing theBurgess bound by Theorem 1.1. We have L ( 12 , χ ) (cid:28) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) n χ ( n ) √ n V ( n √ q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where V is a certain smooth function which satisfies V ( y ) (cid:28) (1 + y ) − and V (cid:48) ( y ) (cid:28) y (1 + y ) − . We have (cid:18) x − V (cid:18) x √ q (cid:19)(cid:19) (cid:48) (cid:28) x − (cid:18) x √ q (cid:19) − so we may apply summation by parts to deduce that L ( 12 , χ ) (cid:28) (cid:90) ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) n ≤ x χ ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − (cid:18) x √ q (cid:19) − dx. By Theorem 1.1 we obtain (cid:88) n ≤ x χ ( n ) (cid:28) (cid:15) x q + O ( δ )+ (cid:15) and therefore L ( 12 , χ ) (cid:28) (cid:15) q + O ( δ )+ (cid:15) (cid:32)(cid:90) √ q x − dx + √ q (cid:90) ∞√ q x − dx (cid:33) (cid:28) q + O ( δ )+ (cid:15) . eferences [1] V. Blomer and D. Mili´cevi´c. The second moment of twisted modular l -functions.arXiv:1404.7845.[2] D. A. Burgess. The distribution of quadratic residues and non-residues. Mathematika ,4:106–112, 1957.[3] D. A. Burgess. On character sums and L -series. Proc. London Math. Soc. (3) , 12:193–206, 1962.[4] D. A. Burgess. On character sums and L -series. II. Proc. London Math. Soc. (3) ,13:524–536, 1963.[5] M.-C. Chang. Short character sums for composite moduli. J. Anal. Math. , 123:1–33,2014.[6] P. Deligne. La conjecture de Weil. II. Inst. Hautes ´Etudes Sci. Publ. Math. , (52):137–252, 1980.[7] ´E. Fouvry, E. Kowalski, and P. Michel. A study in sums of products. Phil. Trans. R.Soc. A. , 2015. to appear.[8] P. X. Gallagher. Primes in progressions to prime-power modulus. Invent. Math. , 16:191–201, 1972.[9] L. Goldmakher. Character sums to smooth moduli are small. Canad. J. Math. ,62(5):1099–1115, 2010.[10] S. W. Graham and G. Kolesnik. van der Corput’s method of exponential sums , volume126 of London Mathematical Society Lecture Note Series . Cambridge University Press,Cambridge, 1991.[11] S. W. Graham and C. J. Ringrose. Lower bounds for least quadratic nonresidues. In Analytic number theory (Allerton Park, IL, 1989) , volume 85 of Progr. Math. , pages269–309. Birkh¨auser Boston, Boston, MA, 1990.[12] D. R. Heath-Brown. Hybrid bounds for Dirichlet L -functions. Invent. Math. , 47(2):149–170, 1978.[13] D. R. Heath-Brown. The largest prime factor of X + 2. Proc. London Math. Soc. (3) ,82(3):554–596, 2001.[14] A. J. Irving. The divisor function in arithmetic progressions to smooth moduli. Int.Math. Res. Not. IMRN , 2014. to appear.[15] H. Iwaniec. On zeros of Dirichlet’s L series. Invent. Math. , 23:97–104, 1974.2216] H. Iwaniec and E. Kowalski. Analytic number theory , volume 53 of American Mathe-matical Society Colloquium Publications . American Mathematical Society, Providence,RI, 2004.[17] D. Mili´cevi´c. Sub-Weyl subconvexity for Dirichlet l-functions to prime power moduli. Compos. Math. , 2014. to appear.[18] A. G. Postnikov. On Dirichlet L -series with the character modulus equal to the powerof a prime number. J. Indian Math. Soc. (N.S.) , 20:217–226, 1956.[19] A. Weil. Sur les courbes alg´ebriques et les vari´et´es qui s’en d´eduisent . Actualit´es Sci.Ind., no. 1041 = Publ. Inst. Math. Univ. Strasbourg (1945). Hermann et Cie., Paris,1948.Centre de recherches math´ematiques,Universit´e de Montr´eal,Pavillon Andr´e-Aisenstadt,2920 Chemin de la tour, Room 5357,Montr´eal (Qu´ebec) H3T 1J4 [email protected] PPENDIX: SUMS OF A. IRVING ´ETIENNE FOUVRY, EMMANUEL KOWALSKI, AND PHILIPPE MICHEL Let p be a prime, h ∈ F × p a fixed invertible element modulo p . Let χ be a fixed non-trivialmultiplicative character modulo p .We consider the function W : F p −→ C defined by W ( x ) = 1 √ p X n ∈ F p χ ( n ) χ ( n + h ) e p ( nx ) . In the notation of A. Irving’s paper, we have W ( x ) = p − / W χ,h ( x ).The goal is to study sums of products of values of W of the type S ( y ) = X x ∈ F p n Y i =1 W ( x + t i ) n Y j =1 W ( x + s j ) e p ( xy )where the t i ’s and s j ’s are in F p and y ∈ F p . The sum K χ,h ( h , . . . , h k , y ) in Irving’s paperis of this type, up to a factor p − k / , with n = n = 2 k − for some integer k > t i ) (resp. ( s j )’s) are the sums X i ∈ I h i where I ranges over I ⊂ { , . . . , k } with | I | even (resp. odd).We will prove: Theorem 1. Assume p > . We have | S ( y ) | (cid:28) √ p where the implied constant depends only on n and n if any of the following conditions hold: (1) If y + ( n − n ) h = 0 ; (2) If y + ( n − n ) h/ and there is some x ∈ F p such that the total multiplicity |{ i | t i = x }| + |{ j | s j = x }| is odd. The proof is based on the methods of algebraic geometry described in [1], based on theRiemann Hypothesis over finite fields of Deligne and additional ideas and computations ofKatz [2]. We begin by a remark that allows us to remove the complex conjugates, and whichis needed to apply cleanly the results of [1]. We define W ( x ) = W ( x ) e p (¯2 hx ) . Date : March 18, 2015, 9:59.Ph. M. was partially supported by the SNF (grant 200021-137488) and the ERC (Advanced ResearchGrant 228304). ´E. F. thanks ETH Z¨urich, EPF Lausanne and the Institut Universitaire de France forfinancial support. Ph.M. and E.K. were partially supported by a DFG-SNF lead agency program grant(grant 200021L 153647). emma 2. We have S ( y ) = e p ( αh ) T (cid:16) y + ( n − n ) h (cid:17) where α = 12 (cid:16)X i t i − X j s j (cid:17) and T ( y ) = X x ∈ F p n + n Y l =1 W ( x + u l ) e p ( xy ) where u l = t l for l n and u l = s l − n for n + 1 l n + n .Proof. This is a straightforward computation where the point is that W ( x ) is real for all x ∈ F p ; this follows in turn from the relation e p ( − hx ) W ( x ) = 1 √ p X n ∈ F p χ ( n ) χ ( n + h ) e p ( − ( n + h ) x )= 1 √ p X m ∈ F p χ ( − m − h ) χ ( − m ) e p ( mx ) = W ( x ) . (cid:3) Hence the result follows from the following result concerning sums like T : Proposition 3. Let ( u , . . . , u n ) be distinct elements of F p , let ν i > . Let T ( y ) = X x ∈ F p n Y i =1 W ( x + u i ) ν i e p ( xy ) for y ∈ F p . Then we have T (cid:28) p / , where the implied constant depends only on n and the ν i , provided either that y = 0 , or that y = 0 and some ν i is odd.Proof. We will first apply [1, Th. 2.7] and then [1, Prop. 1.1] to the following data: • the sheaf G is L ψ ( − yX ) , with ψ the additive character corresponding to e p , • the family of sheaves is ( F i ) i n with F i = [+ u i ] ∗ F , where F = FT ψ ( H ) ⊗ L ψ ( hX/ , with H the sheaf-theoretic Fourier transform of theKummer sheaf L χ ( X )¯ χ ( X + h ) . • the open set U is the complement of {− u i } in the affine line.By definition of the Fourier transform, the trace function of F is the function W (one mustcheck that the tensor product defining H is a middle-extension, but this is straightforward),and that of F i is W ( x + u i ), so the sum T is exactly of the type controlled by [1, Prop. 1.1]in that case.We now check that the family ( F i ) is strictly U -generous, in the sense of [1, Def. 2.1],which is the hypothesis in [1, Th. 2.7]. This is true because: 1) The F i are geometrically irreducible middle-extensions of weight 0, because so is F ,in turn because it is a tensor product of an Artin-Schreier sheaf (of weight 0 and ofrank 1) with the Fourier transform of the middle-extension sheaf H of weight 0 whichis geometrically irreducible and of Fourier type (not being an Artin-Schreier sheaf);as noted above, the middle-extension property is checked directly.(2) The geometric monodromy group of F is equal to SL for p > F i , and this group satisfies the secondcondition of loc. cit.,(3) Any pairs of SL with their standard representations is Goursat-adapted (see [2,Example 1.8.1, p. 25]), so the third condition holds;(4) For i = j , we have no geometric isomorphism F i ’ F j ⊗ L , or D( F i ) ’ F j ⊗ L , of sheaves on U , for any rank 1 sheaf L lisse on U (see also below), where D( F i ) isthe dual of F i .Leaving for a bit later the last checks indicated, we can finish the proof of the propositionfirst. From [1, Prop. 1.1], it is enough to show that the abstract “diagonal” classificationof [1, Th. 2.7] implies that H c ( U × ¯ F p , O F ⊗ ν i i ⊗ D( G )) = 0unless y = 0 and all ν i are even. But [1, Th. 2.7] shows that this cohomology group vanishes unless we have a geometric isomorphism G ’ O Λ i ( F i )where Λ i is an irreducible representation of SL contained in the ν i -th tensor of its standardrepresentation (the covering π in loc. cit. is trivial here because we are in a strictly U -generous situation). Assume we have such an isomorphism. Then, since the rank of G is 1,we see that Λ i is trivial for all i , and therefore G is also geometrically trivial, which meansthat y = 0. Then we see that ν i must be even for the trivial representation to be containedin the ν i -th tensor power. (Note that this is the same argument as for sums of products ofclassical Kloosterman sums in [1, Cor. 3.2]).We now finish checking the conditions (2) and (4) above. For (2), we first need to provethat the connected component of the identity G of the geometric monodromy group G of F is SL . Since the tensor product with L ψ ( hX/ does not alter this property, this is a specialcase of [2, Th. 7.9.4], because H is a tame pseudoreflection sheaf, ramified only at ∞ andat the points in S = { , − h } ⊂ A (see also [2, Th. 7.9.6]). We then need to check that infact G is connected so that G = G = SL . This is because F is geometrically self-dual, asfollows from the formula [2, Th. 7.3.8 (2)] for the dual of a Fourier transform (intuitively,this is because W is real-valued). Indeed, G is semisimple and hence of the form µ N SL forsome N > 1, where µ N is the group of N -th roots of unity. This group is self-dual in thestandard representation if and only if N 2, but µ SL = SL .For (4), assume first that we have a geometric isomorphism(1) F i = [+ u i ] ∗ F ’ [+ u j ] ∗ F ⊗ L = F j ⊗ L (on U ) for some rank 1 sheaf L . y [2, Cor. 7.4.6] (1), the sheaf F is tamely ramified at 0 with drop 1, and it is unramifiedon G m by [2, Th. 7.9.4]. If i = j , then the left-hand sheaf in (1) is therefore unramified at − u j , since u i = u j . On the other hand, F j is ramified at − u j and has rank 2 and drop 1.Thus the tensor product with the rank 1 sheaf L is ramified at − u j (the dimension of theinertial invariants is at most 1), which is a contradiction. Thus an isomorphism as above isimpossible unless i = j .There remains to deal with the possibility of an isomorphism F i = [+ u i ] ∗ F ’ [+ u j ] ∗ D( F ) ⊗ L = D( F j ) ⊗ L . But we have seen that D( F ) ’ F , so this is also impossible if i = j . (cid:3) References [1] ´E. Fouvry, E. Kowalski and Ph. Michel: A study in sums of products , Phil. Trans. R. Soc. A, to appear; dx.doi.org/10.1098/rsta.2014.0309 [2] N.M. Katz: Exponential sums and differential equations , Annals of Math. Studies 124, Princeton Univ.Press (1990). Universit´e Paris Sud, Laboratoire de Math´ematique, Campus d’Orsay, 91405 Orsay Cedex,France E-mail address : [email protected] ETH Z¨urich – D-MATH, R¨amistrasse 101, CH-8092 Z¨urich, Switzerland E-mail address : [email protected] EPFL/SB/IMB/TAN, Station 8, CH-1015 Lausanne, Switzerland E-mail address : [email protected]@epfl.ch