ANNALS OF MATHEMATICS
Weak subconvexity for central values of L-functions
By Kannan Soundararajan
SECOND SERIES, VOL. 172, NO. 2 September, 2010
anmaah Annals of Mathematics, 172 (2010), 1469–1498
Weak subconvexity for central values of L-functions
By KANNAN SOUNDARARAJAN
Abstract
We describe a general method to obtain weak subconvexity bounds for many classes of L-functions. We give several examples of our bound, and our work has applications to a conjecture of Rudnick and Sarnak for the mass equidistribution of Hecke eigenforms
1. Introduction and statement of results
A fundamental problem in number theory is to estimate the values of L- functions at the center of the critical strip. The Langlands program predicts that all L-functions arise from automorphic representations of GL.N / over a number field, and moreover that such L-functions can be decomposed as a product of primitive L-functions arising from irreducible cuspidal representations of GL.n/ over ޑ. The L-functions that we consider will either arise in this manner, or will be the Rankin-Selberg L-function associated to two irreducible cuspidal representations. Note that such Rankin-Selberg L-functions are themselves expected to arise from automorphic representations, but this is not known in general. Given an irreducible cuspidal automorphic representation (normalized to have unitary central character), we denote the associated standard L-function by L.s; /, and its analytic conductor (whose definition we shall recall shortly) by 1 1 " C./. There holds generally a convexity bound of the form L. ; / " C./ 4 2 C (see Molteni [28]).1 The Riemann hypothesis for L.s; / implies the Lindelof¨ hypothesis: L. 1 ; / C./". In several applications it has emerged that the con- 2 vexity bound barely fails to be of use, and that any improvement over the convexity bound would have significant consequences. Obtaining such subconvexity bounds 1 1 ı has been an active area of research, and estimates of the type L. ; / C./ 4 2 The author is partially supported by the National Science Foundation (DMS-0500711) and the Amer- ican Institute of Mathematics (AIM). 1Recently, Roger Heath-Brown [14] has pointed out an elegant application of Jensen’s formula 1 1 for strips that leads generally to the stronger convexity bound L. ; / C./ 4 . 2 1469 1470 KANNAN SOUNDARARAJAN for some ı > 0 have been obtained for several important classes of L-functions. However in general the subconvexity problem remains largely open. For compre- hensive accounts on L-functions and the subconvexity problem we refer to Iwaniec and Sarnak [21], and Michel [27]. In this paper we describe a method that leads in many cases to an improve- ment over the convexity bound for values of L-functions. The improvement is not a saving of a power of the analytic conductor, as desired in formulations of 1 the subconvexity problem. Instead we obtain an estimate of the form L. 2 ; / 1 1 " C./ 4 =.log C.// , which we term weak subconvexity. In some applications, it appears that a suitable weak subconvexity bound would suffice in place of genuine subconvexity. In particular, by combining sieve estimates for the shifted convo- lution problem developed by Holowinsky together with the weak subconvexity estimates developed here, Holowinsky and I [18] have been able to resolve a con- jecture of Rudnick and Sarnak on the mass equidistribution of Hecke eigenforms. We begin by giving three illustrative examples of our work before describing the general result. Example 1. Let f be a holomorphic Hecke eigenform of large weight k for the full modular group SL2.ޚ/. Let t be a fixed real number (for example, t 0), 1 2 D and consider the symmetric square L-function L. 2 it; sym f /. The convexity C 1 1 2 2 " bound for this L-function gives L. 2 it; sym f / t k C , and this can be 1 j C j refined to t k 2 by Heath-Brown’s result mentioned in footnote 1. Our method gives the weak subconvexity bound, for any " > 0,
1 3 2 4 1 2 k .1 t / (1.1) L. 2 it; sym f / " C j1 j" : j C j .log k/ Obtaining subconvexity bounds in this situation (with a power saving in k) remains an important open problem. In the case when k is fixed, and t gets large such a subconvexity bound has been achieved recently by Li [22]. We have assumed that the level is 1 for simplicity, and the result holds for higher level also. However, the assumption that f is holomorphic is essential, since our method makes use of the Ramanujan conjectures known here due to Deligne. Similar results would hold for Maass forms if we assume the Ramanujan conjectures, but unfortunately the partial results known towards the Ramanujan bounds are insufficient for our purpose. Example 2. Let f be a holomorphic Hecke eigenform of large weight k for the full modular group. Let be a fixed Hecke-Maass eigencuspform for SL2.ޚ/. 1 Consider the triple product L-function L. 2 ; f f /. The convexity bound 1 1 " gives L. ; f f / k . Our weak subconvexity bound gives for any 2 C " > 0
1 k (1.2) L. 2 ; f f / ;" 1 " : .log k/ WEAKSUBCONVEXITYFORCENTRALVALUESOF L-FUNCTIONS 1471
Again we could consider higher level, but the assumption that f is holomorphic is necessary for our method.
Example 3. Let 0 be an irreducible cuspidal automorphic representation on GL.m0/ over ޑ with unitary central character. We treat 0 as fixed, and consider 1 1 L. 2 it; 0/ in the t-aspect. The convexity bound here is L. 2 it; 0/ 0 m C 0 " C .1 t / 4 and we obtain C j j C m0 4 1 .1 t / (1.3) L. it; 0/ ;" C j j : j 2 C j 0 .log.1 t //1 " C j j Similarly, if .mod q/ is a primitive Dirichlet character with q large, then
m0 4 1 q (1.4) L. 2 ; 0 / 0;" 1 " : .log q/ The most general example along these lines is the following: Let 0 be as above, and let be an irreducible cuspidal automorphic representation on GL.m/ with unitary central character and such that satisfies the Ramanujan conjectures. Then 1 we would obtain a weak subconvexity bound for L. ; 0 /. 2 We now describe an axiomatic framework (akin to the Selberg class) for the class of L-functions that we consider. The properties of L-functions that we as- sume are mostly standard, and we have adopted this framework in order to clarify the crucial properties needed for our method. In addition to the usual assumptions of a Dirichlet series with a Euler product and a functional equation, we will need an assumption on the size of the Dirichlet series coefficients. We call this a weak Ramanujan hypothesis, as the condition is implied by the Ramanujan conjectures. The reader may prefer to ignore our conditions below and restrict his attention to automorphic L-functions satisfying the Ramanujan conjectures, but our framework allows us to deduce results even in cases where the Ramanujan conjectures are not known. Let m 1 be a fixed natural number. Let2 L.s; / be given by the Dirichlet series and Euler product m X1 a .n/ Y Y ˛j; .p/Á 1 (1.5a) L.s; / 1 ; D ns D ps n 1 p j 1 D D and we suppose that both the series and product are absolutely convergent in Re.s/ > 1.We write m s Y (1.5b) L.s; / N 2 ޒ.s j /; 1 D C j 1 D
2Here the notation is meant to suggest that corresponds to an automorphic representation, but this is not assumed. 1472 KANNAN SOUNDARARAJAN
s=2 where ޒ.s/ .s=2/, N denotes the conductor, and the j are complex D numbers. The completed L-function L.s; /L.s; / has an analytic continua- tion3 to the entire complex plane, and has finite order.1 Moreover, it satisfies a functional equation (1.5c) L.s; /L.s; / L.1 s; /L.1 s; /; 1 D z1 z where is the root number (a complex number of magnitude 1), and m X1 a .n/ s Y (1.5d) L.s; / ; and L.s; / N 2 ޒ.s j /: Q D ns Q 1 D C n 1 j 1 D D We define the analytic conductor C C./ (see [21]) by D m Y (1.5e) C./ N .1 j /: D C j j j 1 D 1 Our goal is to obtain an estimate for L. 2 ; / in terms of the analytic conductor C./. Properties (1.5a)–(1.5d) are standard features of all interesting L-functions. We now need an assumption on the size of the numbers ˛j; .p/. The Ramanujan conjectures, which are expected to hold for all L-functions, predict that ˛j; .p/ 1 j jÄ for all p. Further, it is expected that the numbers j appearing in (1.5b) all satisfy Re.j / 0. Towards the Ramanujan conjectures it is known (see [25]) that if is 1 ım an irreducible cuspidal representation of GL.m/ then ˛j; .p/ p 2 for all p, 1 2 j j Ä and that Re.j / ım where ım 1=.m 1/. We will make the following 2 C D C weak Ramanujan hypothesis. Write
L X1 .n/ƒ.n/ (1.6a) 0 .s; / ; L D ns n 1 D k Pm k where .n/ 0 unless n p is a prime power when it equals j 1 ˛j; .p/ . D D D We assume that for some constants A0;A 1, and all x 1 there holds 2 X .n/ A0 (1.6b) j j ƒ.n/ A2 : n log ex x (1.6c) Re.j / 1 ım; for some ım > 0; and all 1 j m: C Ä Ä 3Thus we are not allowing L.s; / to have any poles. It would not be difficult to modify our results to allow the completed L-function to have poles at 0 and 1. 4This assumption is very weak: from [25] we know that it holds for all automorphic L-functions, and also for the Rankin-Selberg L-function associated to two automorphic representations. WEAKSUBCONVEXITYFORCENTRALVALUESOF L-FUNCTIONS 1473 THEOREM 1. Let L.s; / be an L-function satisfying the properties (1.5a)– (1.5e) and (1.6a)–(1.6c). Then for any " > 0 we have 1 4 1 C./ L. 2 ; / 1 " : .log C.// Here the implied constant depends on m, A, A0, ım, and ". We now show how the examples given above fit into the framework of Theo- rem 1. Example 1 (proof). Write the Euler product for L.s; sym2f / as ˛2 Á 1 Á 1 ˇ2 Á 1 2 Y p ˛pˇp p L.s; sym f / 1 s 1 s 1 s ; D p p p p where ˛p ˇp are complex numbers of magnitude 1 (by Deligne), and ˛p ˇp D C equals the p-th Hecke eigenvalue of f . From the work of Shimura we know that the completed L-function 2 2 ƒ.s; sym f / ޒ.s 1/ޒ.s k 1/ޒ.s k/L.s; sym f / D C C C is entire and satisfies the functional equation ƒ.s; sym2f / ƒ.1 s; sym2f /. D Thus criteria (1.5a)–(1.5d) hold, and the analytic conductor of (1.5e) is k2. If we 2 k write L0=L.s; sym f / in the notation of (1.6a), then the analogous sym2f .p / 2k 2k equals ˛ 1 ˇp which is 3 in magnitude. Thus criterion (1.6b) holds with p C C Ä A 3, and A0 being some absolute constant. Visibly, criterion (1.6c) also holds. D 1 2 1 1 " Therefore Theorem 1 applies and yields L. ; sym f / k 2 =.log k/ . 2 We have shown (1.1) when t 0. To obtain the general case, we apply the D 2 2 framework of Theorem 1 to the shifted function Lt .s; sym f / L.s it; sym f /. WD C The criteria we require hold. The only difference is that we must make correspond- ing shifts to the -functions appearing in the functional equation. These shifts imply that the analytic conductor is now .1 t /.k t /2 k2.1 t /3. C j j C j j C j j Applying Theorem 1, we complete the proof of (1.1). Example 2 (proof). Write the p-th Hecke eigenvalue of f as ˛ .p/ ˇ .p/, f C f where ˛ .p/ˇ .p/ 1 and ˛ .p/ ˇ .p/ 1. Write the p-th Hecke eigen- f f D j f j D j f j D value of as ˛ .p/ ˇ .p/ where ˛ .p/ˇ .p/ 1, but we do not know here C D the Ramanujan conjecture that these are both of size 1. Write also the Laplace 1 2 5 eigenvalue of as t , where t ޒ. D 4 C 2 5This is true since we are working on the full modular group. For a congruence subgroup, we 3 1 could use Selberg’s bound that the least eigenvalue is which gives that Im.t / ; see [25]. 16 j j Ä 4 1474 KANNAN SOUNDARARAJAN The triple product L-function L.s; f f / is then defined by means of the Euler product of degree 8 (absolutely convergent in Re.s/ > 1) ˛ .p/2˛ .p/Á 1 Á 2 ˇ .p/2˛ .p/Á 1 Y f ˛ .p/ f 1 s 1 s 1 s p p p p 2 2 ˛ .p/ ˇ .p/Á 1 ˇ .p/Á 2 ˇ .p/ ˇ .p/Á 1 1 f 1 1 f : ps ps ps This L-function is not primitive and factors as L.s; /L.s; sym2f /. Consider the product of eight -factors L .s; f f / 1 Y ޒ.s k 1 it /ޒ.s k it /ޒ.s it /ޒ.s 1 it /: WD C ˙ C ˙ ˙ C ˙ ˙ From the work of Garrett [6], it is known that L.s; f f /L .s; f f / is an 1 entire function in ރ, and its value at s equals its value at 1 s. Thus criteria (1.5a)– 4 8 (1.5d) are met, and the analytic conductor in (1.5e) is of size k .1 t / . C j j If we write L0 X s L0 X s .s; f / f .n/ƒ.n/n and .s; / .n/ƒ.n/n ; L D n L D n then 2 L0 X f f .n/ƒ.n/ X f .n/ .n/ƒ.n/ .s; f f / s s : L D n n D n n If n pk then .n/ ˛ .p/k ˇ .p/k 2, and so to check (1.6b) for D j f j D j f C f j Ä f f , we need only show that 2 X .n/ ƒ.n/ A0 j j A2 ; n log.ex/ x (a holomorphic or Maass eigencuspform) and allowing f1 and f2 to vary over holomorphic eigencuspforms. Or we could consider f1 f2 f3 with all three varying over holomorphic eigencuspforms. Subconvexity bounds (with a power saving) have been obtained for triple products 1 2 3 where 1 and 2 are considered fixed, and 3 varies over holomorphic or Maass forms, see [1] and [32]. Example 3 (proof). This example follows upon using the ideas in the proof of Examples 1 and 2. The weak Ramanujan hypothesis (1.6b) is verified by an appeal to the Rankin-Selberg theory for L.s; 0 0/ as in Example 2. The gen- z eral Rankin-Selberg theory is the culmination of work by many authors, notably Jacquet, Piatetskii-Shapiro, and Shalika, Shahidi, and Moeglin and Waldspurger; a convenient synopsis of the analytic features of this theory may be found in [30]. A narrow zero-free region (which is sufficient to check (1.6b)) for general Rankin- Selberg L-functions has been established by Brumley [3]. 1 The value L. it; 0/ is bounded by shifting L-functions as in Example 1. 2 C We omit further details. Application to the mass equidistribution of Hecke eigenforms. Perhaps the most interesting application of our results pertains to a conjecture of Rudnick and Sarnak on the mass equidistribution of Hecke eigenforms. For simplicity, consider a holomorphic Hecke eigencuspform f of weight k for the full modular group SL2.ޚ/. Consider the measure D dx dy yk f .z/ 2 ; f D j j y2 where we suppose that f has been normalized to satisfy Z k 2 dx dy y f .z/ 2 1: ވ j j y D n Rudnick and Sarnak ([29]; see also [26], [30]) have conjectured that as k , 3 dxdy ! 1 the measure f approaches the uniform distribution measure y2 on the funda- mental domain X ވ. This is the holomorphic analog of their quantum unique D n ergodicity conjecture for Maass forms. Quantum unique ergodicity for Hecke- Maass forms is now known thanks to the breakthrough work of Lindenstrauss [23] together with a final ingredient (ruling out escape of mass) supplied by Soundarara- jan [31]. However, the ergodic theoretic methods introduced by Lindenstrauss do not seem to apply to the holomorphic case. Set F .z/ yk=2f .z/, and recall the Petersson inner product of two nice k D functions g1 and g2 on X Z dx dy g1; g2 g1.z/g2.z/ 2 : h i D X y A smooth bounded function on X has a spectral expansion in terms of the constant function, the space of Maass cusp forms which are eigenfunctions of the Laplacian 1476 KANNAN SOUNDARARAJAN and all Hecke operators, and the Eisenstein series E.z; 1 it/ with t ޒ; see 2 C 2 Iwaniec [19]. Thus by an analog of Weyl’s equidistribution criterion, the Rudnick-Sarnak conjecture amounts to showing that F ;F ; E. ; 1 it/F ;F 0 h k ki h 2 C k ki ! as k , where is a fixed Maass cusp form which is an eigenfunction of the ! 1 Laplacian and all Hecke operators, and E.z; 1 it/ denotes the Eisenstein series 2 C (with t fixed). Using the unfolding method, it is easy to show that 1 1 2 1 1 ˇ 3 . 2 it/L. 2 it; sym f / .k 2 it/ˇ E. ; it/F ;F ˇ 2 C C C ˇ: jh 2 C k kij D ˇ .1 2it/L.1; sym2f / .k/ ˇ C 1 1 1 1 Since .k it/ .k /, . it/ .1 t / 4 , and .1 2it/ j 2 C j Ä 2 j 2 C j C j j j C j 1= log.1 t /, using Stirling’s formula and our bound (1.1) it follows that C j j 2 1 .1 t / E. ; 2 it/Fk;Fk " 1 "C j j 2 : jh C ij .log k/ L.1; sym f / For the case of a Maass cusp form, a beautiful formula of Watson (see Theo- rem 3 of [33]) shows that (here has been normalized so that ; 1) h i D 1 1 2 1 L . 2 ; f f /L. 2 ; f f / F ;F 1 ; jh k kij D 8 ƒ.1; sym2f /2ƒ.1; sym2 / where L.s; f f / is the triple product L-function of Example 2, and L 1 denotes its Gamma factors (see Example 2 (proof)), and 2 2 ƒ.s; sym f / ޒ.s 1/ޒ.s k 1/ޒ.s k/L.s; sym f /; D C C C and 2 2 ƒ.s; sym / ޒ.s/ޒ.s 2it /ޒ.s 2it /L.s; sym /: D C Using Stirling’s formula and the bound (1.2) of Example 2, we conclude that 1 Fk;Fk 1 : jh ij 2 " 2 .log k/ L.1; sym f / COROLLARY 1. With notation as above, we have 1 Fk;Fk ;" 1 : jh ij 2 " 2 .log k/ L.1; sym f / Moreover, 2 1 .1 t / E. ; 2 it/Fk;Fk " 1 "C j j 2 : jh C ij .log k/ L.1; sym f / Given ı > 0, Corollary 1 shows that if f ranges over those Hecke eigencusp- 2 1 ı forms with L.1; sym f / .log k/ 2 then as k , the measure converges C ! 1 f WEAKSUBCONVEXITYFORCENTRALVALUESOF L-FUNCTIONS 1477 3 dx dy 2 to y2 . This criterion on L.1; sym f / is expected to hold for all eigenforms f ; for example, it is implied by the Riemann hypothesis for L.s; sym2f /. Using large sieve estimates6 one can show that the number of exceptional eigenforms f with weight k K for which the criterion fails is K". Our criterion complements Ä the work of Holowinsky [17], who attacks the mass equidistribution conjecture by an entirely different method. Combining his results with ours gives a complete res- olution of the Rudnick-Sarnak conjecture on mass equidistribution for eigenforms. A detailed account of this result will appear in a joint work with Holowinsky [18]. As noted earlier, the barrier to using our methods for Maass forms is the Ramanujan conjecture which remains open here. The weak Ramanujan hypothesis that we need could be verified (using the large sieve) for all but T " Maass forms with Laplace eigenvalue below T . Thus with at most T " exceptions, one could establish the equidistribution of Maass forms on SL2.ޚ/ ވ. n For simplicity we have confined ourselves to SL2.ޚ/ above. Similar D results apply to congruence subgroups of level N . One could also consider the cocompact case of quaternion division algebras. If the quaternion algebra is un- ramified at infinity, Lindenstrauss’s work shows the equidistribution of Maass cusp forms of large eigenvalue. Our results would show the corresponding equidistribu- tion for holomorphic eigenforms, allowing for a small number of exceptional cases (at most K" exceptions with weight below K). In the case of a ramified quaternion algebra (acting on the unit sphere S 2), the problem concerns the equidistribution of eigenfunctions on the sphere (see [2]) , and again our results establish such equidistribution except for a small number of cases (omitting at most N " spherical harmonics of degree below N that are also Hecke eigenforms). In these compact cases, there is no analog of Holowinsky’s work, and so we are unable to obtain a definitive result as in SL2.ޚ/ ވ. n We now return to the setting of Theorem 1, and describe an auxiliary result which will be used to prove Theorem 1. For an L-function satisfying (1.5a)–(1.5e) and (1.6a)–(1.6c), we may use the convexity bound to establish that (see Lemma 4.2 below)7 X x (1.7) a .n/ ; log x n x Ä 1 B provided x C 2 .log C/ for some positive constant B. Our main idea is to show 1 B that similar cancellation holds even when x C 2 .log C/ for any constant B. D 6Precisely, with very few exceptions one can approximate L.1; sym2f / by a short Euler product. Such results in the context of Dirichlet L-functions are classical, and a detailed account may be found in [7]. In the context of symmetric square L-functions, see Luo [24], and Cogdell and Michel [4]. 7We recall here that L.s; / was assumed not to have any poles. If we alter our framework to allow a pole at s 1, say, then (1.7) would be modified to an asymptotic formula with a main term D of size x. Then Theorem 2 would extrapolate that asymptotic formula to a wider region. 1478 KANNAN SOUNDARARAJAN THEOREM 2. Let L.s; / be as in Theorem 1. For any " > 0, any positive 1 B constant B, and all x C 2 .log C/ we have X x a .n/ : .log x/1 " n x Ä The implied constant may depend on A, A0, m, ım, B and ". Once Theorem 2 is established, Theorem 1 will follow from a standard partial summation argument using an approximate functional equation. In Theorems 1 and 2, by keeping track of the various parameters involved, it would be possible to 1 quantify ". However, the limit of our method would be to obtain a bound C 4 = log C in Theorem 1, and x= log x in Theorem 2. 1 " We have termed the saving of .log C/ as weak subconvexity, and as noted above this is close to the limit of our method. One may legitimately call a saving of any fixed power of log C as weak subconvexity. For example, in the application to the Rudnick-Sarnak conjecture any log power saving together with Holowinsky’s work would suffice to show that F ;F 0, for a fixed Maass cusp form. jh k kij ! However to deal with the Eisenstein series contributions in that application, a sav- ing of a substantial power of log C is needed8, and a small saving would not suffice. Finally, it would be very desirable to establish a version of weak subconvexity 2 ı saving a large power of log C . For example, if one could save .log C/ C for any fixed ı > 0, one would obtain immediately the mass equidistribution conse- quences mentioned above. Similarly it would be desirable to improve upon the weak Ramanujan criterion that we have imposed. 2. Slow oscillation of mean values of multiplicative functions We now discuss the ideas underlying Theorem 2, and state our main technical result from which the results stated in Section 1 follow. At the heart of Theorem 2 is the fact that mean values of multiplicative functions vary slowly. Knowing (1.7) 1 B in the range x C 2 .log C/ , this fact will enable us to extrapolate (1.7) to the 1 B range x C 2 .log C/ . The possibility of obtaining such extrapolations was first considered by Hilde- brand [15], [16]. If f is a multiplicative function, we shall denote by S.x/ D S.x f / the partial sum P f .n/. Hildebrand [16] showed that if 1 f .n/ 1 I n x Ä Ä is a real valued multiplicativeÄ function then for 1 w px Ä Ä 1 1 X w X log x Á 2 Á (2.1) f .n/ f .n/ O log : x D x C log 2w n x n x=w Ä Ä In other words, the mean value of f at x does not change very much from the mean-value at x=w. Hildebrand [15] used this idea to show that from knowing 8A saving of .log C/0:7 would probably be sufficient. WEAKSUBCONVEXITYFORCENTRALVALUESOF L-FUNCTIONS 1479 9 1 " Burgess’s character sum estimates for x q 4 C one may obtain some nontrivial 1 " cancellation even in the range x q 4 . Elliott [5] generalized Hildebrand’s work to cover complex valued multiplica- tive functions with f .n/ 1, and also strengthened the error term in (2.1). Notice j j Ä that a direct extension of (2.1) for complex valued functions is false. Consider f .n/ ni for some real number 0. Then S.x f / x1 i =.1 i / O.1/, D ¤ I D C C C and S.x=w f / .x=w/1 i =.1 i / O.1/. Therefore (2.1) is false, and instead I D C C C we have that S.x/=x is close to wi S.x=w/=.x=w/. Building on the pioneer- ing work of Halasz [11], [12] on mean-values of multiplicative functions, Elliott showed that for a multiplicative function f with f .n/ 1, there exists a real j j Ä number .x/ with log x such that for 1 w px D j j Ä Ä Ä 1 log 2w Á 19 Á (2.2) S.x/ w1 i S.x=w/ O x : D C C log x In [8], Granville and Soundararajan give variants and stronger versions of (2.2), with 1 replaced by 1 2= ". 19 In order to establish Theorem 2, we require similar results when the mul- tiplicative function is no longer constrained to the unit disc. The situation here is considerably more complicated, and instead of showing that a suitable linear combination of S.x/=x and S.x=w/=.x=w/ is small, we will need to consider lin- ear combinations involving several terms S.x=wj /=.x=wj / with j 0; : : : ; J . In D order to motivate our main result, it is helpful to consider two illustrative examples. Example 2.1. Let k be a natural number, and take f .n/ d .n/, the k-th D k divisor function. Then, it is easy to show that S.x/ xP .log x/ O.x1 1=k "/ D k C C where P is a polynomial of degree k 1. If k 2, it follows that S.x/=x k S.x=w/=.x=w/ is of size .log w/.log x/k 2, which is not o.1/. However, if 1 Ä w x1=2k, the linear combination Ä k ! j k ! X j k S.x=w / X j k j 1 1 . 1/ . 1/ P .log x=w / O.x 2k / O.x 2k / j x=wj D j k C D j 0 j 0 D D is very small. Example 2.2. Let 1; : : : ; R be distinct real numbers, and let k1; : : : ; kR be natural numbers. Let f be the multiplicative function defined by F .s/ P s QR kj D n1 1 f .n/n j 1 .s i j / . Consider here the linear combination (for D D D 1 w x1=.2.k1 kR//) Ä Ä CC k1 kR ! 1 X X j1 jR k1 . 1/ CC x j1 j1 0 jR 0 D D ! Á kR j1.1 i 1/ jR.1 i R/ x w C CC C S : j1 jR jR w CC 9For simplicity, suppose that q is cube-free. 1480 KANNAN SOUNDARARAJAN By Perron’s formula we may express this as, for c > 1, R 1 Z c i ds C 1 Y kj 1 i j s kj s 1 .s i j / .1 w / x : 2i C s c i j 1 1 D Notice that the poles of the zeta-functions at 1 i j have been canceled by the C factors .1 w1 i j s/kj . Thus the integrand has a pole only at s 0, and a C D standard contour shift argument shows that this integral is x ı for some ı > 0. Fortunately, it turns out that Example 2.2 captures the behavior of mean-values of the multiplicative functions of interest to us. In order to state our result, we require some notation. Let f denote a multiplicative function and recall that X (2.3) S.x/ S.x f / f .n/: D I D n x Ä We shall write X1 f .n/ (2.4) F .s/ ; D ns n 1 D and we shall assume that this series converges absolutely in Re.s/ > 1. Moreover, we write F X1 f .n/ƒ.n/ X1 ƒf .n/ (2.5) 0 .s/ ; F D ns D ns n 1 n 1 D D where .n/ ƒ .n/ 0 unless n is the power of a prime p. We next assume the f D f D analog of the weak Ramanujan hypothesis (1.6b). Namely, we suppose that there exist constants A, A0 1 such that for all x 1 we have 2 X f .n/ ƒ.n/ A0 (2.6) j j A2 : n log.ex/ x ᏻ .x; w/ ᏻ .x; w 1; : : : ; R/ ` D ` I ! Á X j1 jR ` j1.1 i 1/ jR.1 i R/ x . 1/ CC w C CC C S : D j wj1 jR j ` CC Ä WEAKSUBCONVEXITYFORCENTRALVALUESOF L-FUNCTIONS 1481 THEOREM 2.1. Keep in mind the conditions and notation (2.3) through (2.8). Let X 10 and 1 " > 0 be given. Let R Œ10A2="2 1 and put L Œ10AR, D C 1 D and L .L; : : : ; L/. Let w be such that 0 log w .log X/ 3R . There exist real D Ä 2 Ä numbers 1; : : : ; R with j exp..log log X/ / such that for all 2 x X we j j Ä Ä Ä have x " ᏻL.x; w 1; : : : ; R/ .log X/ : j I j log x The implied constant above depends on A, A0 and ". For a general multiplicative function, we cannot hope for any better bound for the oscillation than x= log x. To see this, suppose w 2, and consider the multiplicative function f with f .n/ 0 for n x=2 and f .p/ 1 for primes D Ä D x=2 < p x. Then S.x/ x= log x whereas S.x=wj / 1 for all j 1, and Ä D therefore for any choice of the numbers 1; : : : ; R we would have ᏻL.x; w/ x= log x. Our proof of Theorem 2.1 builds both on the techniques of Halasz (as de- veloped in [5] and [8]), and also the idea of pretentious multiplicative functions developed by Granville and Soundararajan (see [10] and [9]). In Section 5 we will describe the choice of the numbers 1; : : : ; R appearing in Theorem 2.1, and develop relevant estimates for the Dirichlet series F .s/. Then the proof of Theorem 2.1 will be completed in Section 6. 3. Some preliminary lemmas We collect together here some lemmas that will be useful below. We begin with a combinatorial lemma. LEMMA 3.1. Let b.1/, b.2/; : : : be a sequence of complex numbers. Define the sequence a.0/ 1, a.1/, a.2/; : : : by means of the formal identity D X1 b.k/ Á X1 (3.1) exp xk a.n/xn: k D k 1 n 0 D D For j 1 or 2, define the sequences Aj .0/ 1, Aj .1/, Aj .2/; : : : by means of the D D formal identity j X1 b.k/ kÁ X1 n exp j j x Aj .n/x : k D k 1 n 0 D D j Then Aj .n/ a.n/ for all n. j j Proof. If we expand out the left-hand side of (3.1) and equate coefficients we obtain that X (3.2) a.n/ b.1/ b.r /W./; D 1 r n 1 CC2 Dr 1 1482 KANNAN SOUNDARARAJAN where the sum is over all partitions of n, and W./ > 0 is a weight attached to each partition which is independent of the sequence b.k/. Although we can write down explicitly what the weight W./ is, we do not require this. All we need is that X W./ 1; D P k which follows from (3.2) by taking b.k/ 1 for all k so that exp. k1 1 b.k/x =k/ P n D D 1=.1 x/ 1 x whence a.n/ 1. When j 1 the lemma follows by D D n 0 D D the triangle inequality.D In the case j 2, Cauchy-Schwarz gives D 2 X 2 Á X Á a.n/ b.1/ b.r / W./ W./ j j Ä j j X 2 Á b.1/ b.r / W./ A2.n/: D j j D The significance of Lemma 3.1 for us is the following. Let f be a multiplica- tive function with the Euler factor at p being (compare (1.6a) and (2.5)) n k X1 f .p / X1 f .p /Á exp pns D kpks I n 0 k 1 D D then the lemma guarantees that if we write k 2 .2/ n X1 f .p / Á X1 f .p / exp j j ; kpks D pns k 1 n 0 D D and use this to define a multiplicative function f .2/.n/, then f .n/ 2 f .2/.n/. j j Ä In the case that f .n/ a .n/ corresponds to the coefficients of an automorphic D 10 L-function , this means that the coefficient a .n/ of the Rankin-Selberg 2 Q L-function exceeds a .n/ . The reader may compare this lemma with a similar j j result of Molteni, [28, Prop. 6]. Although we are only interested in Lemma 3.1 in the cases j 1 and 2, an application of Holder’s¨ inequality shows that a similar D result holds for all j 1. LEMMA 3.2. Let f be a multiplicative function, and keep the notation (2.4) and (2.5), and suppose that the criterion (2.6) holds. For all x 2 we have 2 X f .n/ X f .n/ 2 j j .log x/A; and j j .log x/A : n n n x n x Ä Ä Moreover, for all 2 > 1 we have 1 ÁA 1 ÁA 1 F . it/ and F . it/ C : j C j 1 j 0 C j 1 The implied constants may depend on A and A0. 10We assume that n is the power of an unramified prime. WEAKSUBCONVEXITYFORCENTRALVALUESOF L-FUNCTIONS 1483 Proof. These are simple consequences of our weak Ramanujan assumption (2.6). By splitting the sum into intervals ek < n ek 1 and using (2.6), we see Ä C that for any 2 > 1 2 X1 f .n/ ƒ.n/ 1 Á (3.3) j j A2 log O.1/; n log n Ä 1 C n 2 D where the error term above depends on A and A0. By Cauchy-Schwarz it follows also that X1 f .n/ ƒ.n/ 1 Á (3.4) j j A log O.1/: n log n Ä 1 C n 2 D Using Lemma 3.1 and (3.4) we see that X f .n/ X1 f .n/ X f .n/ ƒ.n/ Á j j j j exp j j .log x/A: n n1 1= log.ex/ Ä n1 1= log.ex/ log n n x n 1 C n 2 C Ä D This proves our first inequality. The second inequality follows in the same way, using (3.3) instead of (3.4). The third inequality follows easily from (3.4). Finally, for 2 > 1, ˇF ˇ X f .n/ ƒ.n/ 1 ˇ 0 . it/ˇ j j ˇ F C ˇ Ä n 1 n 2 using Cauchy-Schwarz and (2.6) as in the proof of (3.4). The lemma follows. LEMMA 3.3. Let f be a multiplicative function as in Lemma 3.2. Then for all x 2 X f .n/ x.log x/A: n x j j Ä Moreover, for 1 y x we have Ä Ä X 1 A2=2 f .n/ .yx/ 2 .log x/ : j j x 4. Deduction of the main results from Theorem 2.1 In this section we shall show how Theorems 1 and 2 follow from Theorem 2.1. We begin with Theorem 2, whose proof will require the following simple convexity bound for our L-functions. 1484 KANNAN SOUNDARARAJAN LEMMA 4.1. Let L.s; / be an L-function satisfying the properties (1.5a)– (1.5e) and (1.6a)–(1.6c). Then for all t ޒ we have 2 1 1 m 1 A L. it; / C./ 4 .1 t / 4 .log C.// : j 2 C j C j j C .s 1 it/2 Proof. Define ƒ.s/ L.s; /L.s; /e 2 . Using the Phragmen- D 1 Lindelof¨ principle we may bound ƒ. 1 it/ by the maximum value taken by j 2 C j ƒ.s/ on the lines Re.s/ 1 1= log C , and Re.s/ 1= log C . The functional j j D C D equation shows that the maximum on the line Re.s/ 1= log C is the same as D the maximum on the line Re.s/ 1 1= log C . Therefore, using Lemma 3.2, D C ƒ.1 1= log C it iy/ L. 1 it; / max j C C C j 2 y ޒ 1 j C j Ä L. 2 it; / 2 j C 1 j A y2 L.1 1= log C it iy; / .log C/ max e j C C C 1 j: y ޒ 1 L. 2 it; / 2 j C 1 j Using Stirling’s formula, we may show that ˇޒ.1 1= log C it iy j /ˇ 1 1 ˇ ˇ e2 y .1 t / 4 2 log C ; ˇ C 1 C C C ˇ j j j C ޒ. it j / C j j C j j 2 C C where we used that Re.j / 1 ım to ensure that the numerator stays away C from poles of the -function. The lemma follows immediately. Our next lemma establishes the result (1.7) stated in the introduction. LEMMA 4.2. Let L.s; / be as above. In the range 1 50mA2 x x0 C./ 2 .log C.// WD we have X x a .n/ : log x n x Ä Proof. Observe that for any c > 0, y > 0, any > 0, and any natural number K, 1 Z c i ys es 1ÁK C 1 ds 2i c i s s 1 Z Z Z c i 1 1 C 1 x1 xK s ds K ::: .ye CC / dx1 : : : dxK D 0 0 2i c i s 1 81 if y 1 <ˆ Œ0; 1 if 1 > y e K D 2 :ˆ K 0 if y < e : Therefore, for any c > 1, (4.1) Z c i s s K 1 C 1 x e 1Á X X Á L.s; / ds a .n/ O a .n/ : 2i c i s s D C j j 1 n x x 2 A2 We shall take here K Œm=4 3 and .log x/ . Then for large x we K D C D have e 2, and so applying Lemma 3.3 to the multiplicative function a .n/ we Ä see that the error term above is K 1 A2=2 x (4.2) .e 1/ 2 x.log x/ : log x Now we move the line of integration in the left-hand side of (4.1) to the line c 1=2. Using Lemma 4.1 we see that the integral on the 1=2 line is D 1 1 Z dt 4 2 K A 1 m=4 1 C./ x .log C/ .1 t / C K 1 C j j .1 t / C 1 C j j 1 1 12mA2 C./ 4 x 2 .log.Cx// : Combining this with (4.1) and (4.2) we conclude that for 1 50mA2 x x0 C./ 2 .log C.// WD we have X x a .n/ ; log x n x Ä proving our lemma. Proof of Theorem 2. To prove Theorem 2, we invoke Theorem 2.1. Let R 2 2 D Œ10A =" 1 and L Œ10AR be as in Theorem 2.1. Let x0 be as in Lemma C D1 B LR 4.2, and let x0 x C 2 =.log C/ . Take w x0=x and X xw . Applying D D Theorem 2.1 to the multiplicative function a (note that (1.6b) gives the assumption (2.6)) we find that for an appropriate choice of 1; : : : ; R that X (4.3) ᏻL.X; w/ 1 " : j j .log X/ But, by definition, the left-hand side above is LR 1 LRˇ X ˇ X j ˇ X ˇÁ (4.4) w ˇ a .n/ˇ O w ˇ a .n/ˇ : ˇ ˇ C ˇ ˇ n X=wLR j 0 n X=wj Ä D Ä LR j Now X=w x, and for 0 j LR 1 we have X=w xw x0 so that the D Ä Ä D bound of Lemma 4.2 applies. Therefore (4.4) equals LRˇ X ˇ X Á w ˇ a .n/ˇ O ; ˇ ˇ log X n x C Ä From (4.3) we conclude that ˇ X ˇ LR X x ˇ a .n/ˇ w ; ˇ ˇ .log X/1 " .log x/1 " n x Ä which proves Theorem 2. 1486 KANNAN SOUNDARARAJAN Deduction of Theorem 1 from Theorem 2. Theorem 1 follows from Theorem 2 1 by a standard argument using an “approximate functional equation” for L. 2 ; / (see for example Harcos [13, Th. 2.1]) and partial summation. For the sake of 1 completeness we provide a brief argument. We start with, for c > 2 , Z c i 1 1 C 1 1 L.s 2 ; / s2 ds L.s ; / C 1 e ; 2i 2 1 s c i C L. 2 ; / 1 1 and move the line of integration to Re.s/ c. We encounter a pole at s 0, and D D so the above equals Z c i 1 1 1 C 1 1 L.s 2 ; / s2 ds L. ; / L.s ; / C 1 e : 2 2i 2 1 s C c i C L. 2 ; / 1 1 Now we use the functional equation above, and make a change of variables s s. ! In this way we obtain that Z c i 1 1 1 C 1 1 L.s 2 ; / s2 ds L. ; / L.s ; / C 1 e 2 2i 2 1 s D c i C L. 2 ; / 1 1 Z c i 1 C 1 1 L.s 2 ; / s2 ds L.s ; / C z1 e : 2i 2 1 s C c i C z L. 2 ; / 1 1 Consider the first integral above; the second is estimated similarly. Using Z 1 1 1 X dx L.s 2 ; / .s 2 / a .n/ 3 ; C D C s 2 1 n x x C Ä we see that the first integral above equals Z Z c i 1 1 X 1 C 1 1 L.s 2 ; / s2 s ds Á dx (4.5) a .n/ .s / C 1 e x : 2i 2 1 s 3 1 n x c i C L. 2 ; / x 2 Ä 1 1 To estimate the inner integral over s, we move the line of integration either to 1 ım Re.s/ 2 2 , or to Re.s/ 2. Using Stirling’s formula, we see that this inner D Dı 1 m 2 integral is min..pC =x/ 2 2 ;.pC =x/ /. Thus (4.5) is Z pC Z 1 ım ˇ X ˇ dx 1 ˇ X ˇdx C 4 4 ˇ a .n/ˇ C ˇ a .n/ˇ : ˇ ˇ ım ˇ ˇ 7 1 2 2 C pC 2 n x x n x x Ä Ä We now split into the ranges x pC =.log C/4A=ım and x > pC =.log C/4A=ım . Ä P A In the first range we use Lemma 3.3 to bound n x a .n/ by x.log x/ , and P j Ä 1 " j in the second range we use a .n/ x=.log x/ by Theorem 2. Inserting n x Ä 1 1 " these bounds above, we conclude that the quantity in (4.5) is C 4 =.log C/ , and Theorem 1 follows. WEAKSUBCONVEXITYFORCENTRALVALUESOF L-FUNCTIONS 1487 5. Successive maxima Recall the conditions and notation (2.3) through (2.8). As in Theorem 2.1, X 10 and 1 " > 0 are given, and R Œ10A2="2 1. In this section we D C define the points 1; : : : ; R appearing in Theorem 2.1, and collect together some estimates for the Dirichlet series F .s/. 2 From now on, we shall write T exp..log log X/ /. We define 1 as that point D t in the compact set Ꮿ1 Œ T; T where the maximum of F .1 1= log X it/ D 1 j C 1 C j is attained. Now remove the interval . 1 .log X/ R ; 1 .log X/ R / from C Ꮿ1 Œ T; T , and let Ꮿ2 denote the remaining compact set. We define 2 to be that D point t in Ꮿ2 where the maximum of F .1 1= log X it/ is attained. Next remove 1 j C 1 C j the interval . 2 .log X/ R ; 2 .log X/ R / from Ꮿ2 leaving behind the compact C set Ꮿ3. Define 3 to be the point where the maximum of F .1 1= log X it/ j C C j for t Ꮿ3 is attained. We proceed in this manner, defining the successive maxima 2 1; : : : ; R, and the nested compact sets Ꮿ1 Ꮿ2 ᏯR. Notice that all the points 1; : : : ; R lie in Œ T; T , and moreover are well-spaced: j k 1 j j .log X/ R for j k. ¤ Lemma 3.2 bounds F .1 1= log X it/ by .log X/A. For t Œ T; T j C C j 2 we will show that a much better bound holds, unless t happens to be near one of the points 1; : : : ; R. The next lemma is inspired by the ideas in [10] and [9]. LEMMA 5.1. Let 1 j R and let t be a point in Ꮿj . Then Ä Ä F .1 1= log X it/ .log X/Ap1=j .j 1/=.jR/: j C C j C "=2 In particular, if t ᏯR we have F .1 1= log X it/ .log X/ . 2 j C C j Proof. If t Ꮿj , then for all 1 r j 2 Ä Ä F .1 1= log X it/ F .1 1= log X i j / F .1 1= log X i r / : j C C j Ä j C C j Ä j C C j Therefore, j 1 Y Á j F .1 1= log X i j / F .1 1= log X i r / j C C j Ä j C C j r 1 D 1 .n/ƒ.n/ Á X f i 1 i j exp Re .n n / : Ä j n1 1= log X .log n/ C C n 2 C By Cauchy-Schwarz j .n/ ƒ.n/ ˇ ˇ X f X i r j j ˇ n ˇ n1 1= log X log nˇ ˇ n 2 C r 1 D 2 1 j 1 .n/ ƒ.n/ Á 2 ˇ ˇ2Á 2 X f X ƒ.n/ X i r j j ˇ n ˇ : Ä n1 1= log X log n n1 1= log X log nˇ ˇ n 2 C n 2 C r 1 D 1488 KANNAN SOUNDARARAJAN 2 1 By (2.6) the first factor above is .A log log X O.1// 2 . To handle the second Ä C factor, we expand out the square and obtain j ˇ ˇ2 X ƒ.n/ X i r ˇ n ˇ n1 1= log X log nˇ ˇ n 2 C r 1 D X ƒ.n/ X X ƒ.n/ j 2Re D n1 1= log X log n C n1 1= log X i. r s / log n n 2 C r LEMMA 5.2. Let 1 1= log X. Then C 1 max Ᏺ`. it/ max Ᏺ`.1 1= log X it/ O..log X/ /: t T=2 j C j Ä t T j C C j C j jÄ j jÄ Proof. This follows from the argument of Lemma 2.2 in Granville and Sound- ararajan [8]. For completeness we give a proof. Put 1 1= log X ˛, and assume ˛ Dz C R C ˛ z iz that ˛ > 0. The Fourier transform of k.z/ e j j is k./O 1 e j j dz 2˛ D D 1 D ˛2 2 . By Fourier inversion, we have for any z 1 C 1 Z z ˛ k.log z/ k. log z/ 1 k./z i d D D D 2 O Z T=2 1 1 ˛ i ˛ Á 2 2 z d O : D T=2 ˛ C T C Using this relation appropriately we obtain that 1 Z T=2 ˛ Ᏺ`. it/l 2 2 Ᏺ`.1 1= log X it i/d C D T=2 ˛ C C C C ˛ X1 f .n/ Á O j j : C T n1 1= log X n 1 C D WEAKSUBCONVEXITYFORCENTRALVALUESOF L-FUNCTIONS 1489 Using Lemma 3.2 and partial summation, the error term above is O..log X/A=T / D O..log X/ 1/. If t T=2 then t T , and so j j Ä j C j Ä max Ᏺ`. it/ t T=2 j C j j jÄ Z T=2 1 ˛ 1 max Ᏺ`.1 1= log X iy/ 2 2 d O..log X/ /; Ä y T j C C j T=2 ˛ C j jÄ C and the lemma follows. LEMMA 5.3. Suppose `j L 1 for all 1 j R where we recall that Ä Ä L Œ10AR. Then provided 0 log w .log X/1=.3R/ we have D Ä Ä "=2 max Ᏺ`.1 1= log X it/ .log X/ : t T j C C j j jÄ 1=R Proof. Suppose first that t T but t j > .log X/ for all 1 j j Ä j j Ä j R. Then Lemma 5.1 gives that F .1 1= log X it/ .log X/"=2 and so Ä "=2j C C j Ᏺ`.1 1= log X it/ .log X/ as well. j C C j 1=R Now suppose that t j .log X/ for some 1 j R. By Lemma j j Ä Ä Ä 3.2 we have that F .1 1= log X it/ .log X/A. Moreover, j C C j log w Á`j 2.L 1/ 1= log X it i j `j 3R A 1 w C 1 .log X/ .log X/ ; j j .log X/ R Ä and hence Ᏺ .1 1= log X it/ 1. The lemma follows. j ` C C j Using our work so far, we can record a preliminary estimate for the oscillation which we shall refine in the next section to obtain Theorem 2.1. PROPOSITION 5.4. Suppose `j L 1 for all 1 j R, and that 0 log w 1 Ä Ä Ä Ä .log X/ 3R . For x X we have Ä ᏻ .x; w/ x.log X/2"=3: ` Proof. Since S.x/ x.log x/A by Lemma 3.3, we may assume that log x .log X/"=.2A/, and in particular x w2RL is large. By Perron’s formula we have that for c > 1 1 Z c i es=pT 1Áds X X Á C 1 F .s/zs f .n/ O f .n/ : 2i c i s=pT s D C j j 1 n z z We choose c 1 1= log X and split the integral into two parts: when Im.s/ T D C j j Ä and when Im.s/ > T . For the first range we use Lemma 5.3, and so this portion j j of the integral contributes Z ds x.log X/"=2 j j x.log X/2"=3: Im.s/ T s j jÄ j j For the second region we use that F .s/ .log X/A and deduce that this integral j j contributes Z pT ds x.log X/A j j x: Im.s/ >T s s j j j j j j The proposition follows. 6. Proof of Theorem 2.1 In this section we shall prove Theorem 2.1, with the points 1; : : : ; R being the successive maxima defined in Section 5. Recall the conditions and notation (2.3) through (2.8), and the notation introduced in Section 5. Recall that R D Œ10A2="2 1, that L Œ10AR and L .L; : : : ; L/. Throughout we assume C 1 D D that log w .log X/ 3R , that x X, and all implicit constants will be allowed to Ä Ä depend on A, A0 and ". LEMMA 6.1. With the above notation, we have X " .log x/ᏻL.x; w/ ƒ .d/ᏻL.x=d; w/ O.x.log X/ /: D f C d x Ä j1 jR Proof. Write log x log.x=w / .j1 jR/ log w. Hence, we D CC C C C may express .log x/ᏻL.x; w/ as ! L X j1 jR j1 jR j1 jR j1.1 i 1/ jR.1 i R/ . 1/ CC log.x=w CC /S.x=w CC /w C CC C j L Ä ! L X j1 jR j1 jR j1.1 i 1/ jR.1 i R/ log w . 1/ CC .j1 jR/ S.x=w CC /w C CC C : C C C j j L Ä The second term above is readily seen to be R X 1 i k (6.1) Lw C log w ᏻL ek .x=w; w/; k 1 D where we let ek denote the vector with 1 in the k-th place and 0 elsewhere. Note that the coordinates of L e are all at least L 1, and so by Proposition 5.4, the k quantity in (6.1) is x.log X/". WEAKSUBCONVEXITYFORCENTRALVALUESOF L-FUNCTIONS 1491 To analyze the first term, we write X X .log x/S.x/ f .n/ log n f .n/ log.x=n/ D n x C n x Ä Ä X X Z x dt ƒf .d/f .n=d/ S.t/ D C 1 t n x d n Ä j X Z x dt ƒf .d/S.x=d/ S.t/ : D C 1 t d x Ä Therefore the first term equals X Z x dt ƒf .d/ᏻL.x=d; w/ ᏻL.t; w/ ; C 1 t d x Ä j where we used that R x=w S.t/dt=t R x S.t=wj /dt=t. By Proposition 5.4, the 1 D 1 integral above is x.log X/". The lemma follows. LEMMA 6.2. For 1 z y with y z X we have Ä Ä C Ä LR ˇ 2 2ˇ " X j X ˇ ᏻL.y; w/ ᏻL.y z; w/ ˇ y.log X/ w f .n/ : ˇj j j C j ˇ j j j 0 y=wj It remains to estimate the contribution of the terms d > D. Define temporarily the function g.t/ t log.ex=t/ for 1 t x. By Cauchy-Schwarz we have D Ä Ä X (6.2) ƒ .d/ ᏻL.x=d; w/ j f jj j D X 2 (6.3) g.d/ƒ.d/ ᏻL.x=d; w/ j j D X 2 2Á (6.4) 0.d/ g.d/ ᏻL.x=d; w/ g.d 1/ ᏻL.x=.d 1/; w/ j j C j C j D X d ˇ 2 2ˇ g.d/ˇ ᏻL.x=d; w/ ᏻL.x=.d 1/; w/ ˇ .log X/A 2 ˇj j j C j ˇ D We now turn to the first term in (6.3). For any D < d x and d 1 t d Ä Ä Ä we have that g.d/ g.t/ O.log x/, and by Lemma 6.2 that D C 2 2 ᏻL.x=d; w/ ᏻL.x=t; w/ j j D j j LR x X X Á O .log X/" wj f .n/ : C d j j j 0 x=.dwj / Z d 2 2 2 x 1 2"Á g.d/ ᏻL.x=d; w/ g.t/ ᏻL.x=t; w/ dt O 2 .log X/ C j j D d 1 j j C d LR X X Á O x.log X/1 " wj f .n/ : C C j j j 0 x=.dwj / LR x2 X X .log X/1 2" x.log X/1 " wj f .n/ D C C C j j j 0 n x=.Dwj / D Ä LR x2 X X f .n/ x2 .log X/1 " j j x2: C D C n j 0 n x=.Dwj / D Ä The proof of the proposition is complete. We must now analyze the integral appearing in Proposition 6.3. To this end, we write X S.x/ f .n/ log n; z D n x Ä and define ! X ` ᏻ .x; w/ . 1/j1 jR wj1.1 i 1/ jR.1 i R/S.x=wj1 jR /: z` D CC j C CC C z CC j ` Ä LEMMA 6.4. We have Z x 1 Z x 1 2 dt Á 2 2 dt Á 2 7"=8 ᏻL.t; w/ log.et/ 3 ᏻL.t; w/ 3 .log X/ : 1 j j t 1 jz j t log.et/ C 1494 KANNAN SOUNDARARAJAN Proof. We start as in the proof of Lemma 6.1. Thus we may write R X 1 i k .log t/ᏻL.t; w/ Lw C .log w/ᏻL ek .t=w; w/ D k 1 D ! L X j1 jR j1 jR j1 jR j1.1 i 1/ jR.1 i R/ . 1/ CC log.t=w CC /S.t=w CC /w C CC C : C j j L Ä Since X Z z S.y/ .log z/S.z/ S.z/ f .n/ log.z=n/ S.z/ dy; z z y D C n z D C 1 Ä the second term above may be written as t Z ᏻL.y; w/ ᏻL.t; w/ dy: z C 1 y Putting these remarks together, and using Proposition 5.4 we conclude that 2"=3 .log t/ᏻL.t; w/ ᏻL.t; w/ O.t.1 log w/.log X/ / D z C C 5"=6 ᏻL.t; w/ O.t.log X/ /: D z C The lemma follows. Putting together Proposition 6.3 and Lemma 6.4, we have that x 1 x 1 Z dt Á 2 x 2 2 " ᏻL.x; w/ .log log x/ ᏻL.t; w/ 3 .log X/ : j j log x 1 jz j t log.et/ C log x Theorem 2.1 will now follow from the following proposition. PROPOSITION 6.5. We have Z x 2 dt 3"=2 ᏻL.t; w/ 3 .log X/ : 1 jz j t log.et/ Proof. We make the substitution t ey, obtaining D (6.5) Z x Z log x 2 dt y 2 2y dy ᏻL.t; w/ 3 ᏻL.e ; w/ e 1 jz j t log.et/ D 0 jz j 1 y Z Z C 1 2˛ 1 y 2 2y.1 ˛/ e ᏻL.e ; w/ e C dy d˛: 1= log X 0 jz j The idea now is to estimate the integral over y in (6.5) using Plancherel’s formula. WEAKSUBCONVEXITYFORCENTRALVALUESOF L-FUNCTIONS 1495 Note that the Fourier transform of .ey; w/e y.1 ˛/ is ᏻzL C Z 1 .ey; w/e y.1 ˛ it/dy ᏻzL C C 1 ! X L . 1/j1 jR wj1.1 i 1/ jR.1 i R/ D CC j C CC C j L Ä Z X 1 f .n/ log n e y.1 ˛ it/dy C C y j1 jR 1 n e =w CC Ä ! X L . 1/j1 jR wj1.i 1 ˛ it/ jR.i R ˛ it/ D CC j CC j L Ä X1 f .n/ log n Z 1 e y.1 ˛ it/dy n1 ˛ it C C n 1 C C 0 D R 1 Y .1 w ˛ it i k /LF .1 ˛ it/: D .1 ˛ it/ C 0 C C C C k 1 D Therefore, by Plancherel’s formula we have that Z 1 y 2 y.2 2˛/ (6.6) ᏻL.e ; w/ e C dy 0 jz j R Z Y ˇ ˇ2L dt 1 F .1 ˛ it/ 2 ˇ1 w ˛ it i k ˇ : j 0 C C j ˇ C ˇ 1 ˛ it 2 1 k 1 j C C j D We split the integral in (6.6) into the regions when t T=2 and when t > j j Ä j j T=2. For the latter region we use Lemma 3.2 which gives F .1 ˛ it/ j 0 C C j .log X/A 1 (note that ˛ 1= log X in (6.5)), so that this integral contributes C Z 2A 2 dt .log X/ C 2 1: t >T=2 1 ˛ it j j j C C j For the first region we use Lemmas 5.2 and 5.3 to obtain that R ˇF ˇ Y ˛ it i k L 0 F .1 ˛ it/ 1 w ˇ .1 ˛ it/ˇ ᏲL.1 ˛ it/ j 0 C C j j C j ˇ F C C ˇj C C j k 1 D ˇF ˇ ˇ 0 .1 ˛ it/ˇ.log X/"=2: ˇ F C C ˇ Therefore the integral over the first region contributes Z ˇF ˇ2 dt (6.7) .log X/" ˇ 0 .1 ˛ it/ˇ ˇ ˇ 2 t T=2 F C C 1 ˛ it j jÄ j C C j Z ˇF ˇ2 dt .log X/" 1 ˇ 0 .1 ˛ it/ˇ : ˇ F C C ˇ 1 ˛ it 2 1 j C C j 1496 KANNAN SOUNDARARAJAN y.1 ˛/ P Now the Fourier transform of the function e C n ey ƒf .n/ is Ä Z X X ƒf .n/ 1 1 ƒ .n/e y.1 ˛ it/dy f C C D n1 ˛ it .1 ˛ it/ n ey n C C 1 Ä C C F 1 0 .1 ˛ it/ ; D F C C .1 ˛ it/ C C and so using Plancherel once again we obtain that the quantity in (6.7) is Z ˇ X ˇ2 .log X/" 1 ˇ ƒ .n/ˇ e .2 2˛/ydy: ˇ f ˇ C 0 n ey Ä We conclude that (6.8) Z Z 2 1 y 2 y.2 2˛/ " 1 ˇ X ˇ .2 2˛/y ᏻL.e ; w/ e dy 1 .log X/ ˇ ƒ .n/ˇ e dy: jz j C C ˇ f ˇ C 0 0 n ey Ä Injecting the bound (6.8) in (6.5) we obtain that Z x 2 dt ᏻL.t; w/ 3 1 jz j t log.et/ Z Z ˇ X ˇ2 1 .log X/" 1 e 2˛ 1 ˇ ƒ .n/ˇ e .2 2˛/ydy d˛: C ˇ f ˇ C 1= log X 0 n ey Ä Expanding, we obtain that the double integrals above are Z Z X 1 1 .2 2˛/y 2˛ ƒf .n1/ƒf .n2/ e C dy d˛ j j 1= log X log n2 2 n1 n2 Ä Ä X ƒ.n1/ƒ.n2/ .n1/ .n2/ j f f j 2 2= log X 2 n1 n2 n2C log n2 Ä Ä X 2 2 ƒ.n1/ƒ.n2/ . .n1/ .n2/ / j f j C j f j 2 2= log X 2 n1 n2 n2C log n2 Ä Ä 2 X f .n/ ƒ.n/ j j .log log X/; n1 2= log X log n 2 n C Ä upon using the prime number theorem in the penultimate step, and (2.6) for the last step. 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MR 2618394 (Received September 9, 2008) E-mail address: [email protected] STANFORD UNIVERSITY,DEPARTMENT OF MATHEMATICS, 450 SERRA MALL,BUILDING 380, STANFORD CA 94305-2125, UNITED STATES, http://math.stanford.edu/~ksound/ ISSN 0003-486X ANNALSOF MATHEMATICS This periodical is published bimonthly by the Department of Mathematics at Princeton University with the cooperation of the Institute for Advanced Study. Annals is typeset in TEX by Sarah R. Warren and produced by Mathematical Sciences Publishers. The six numbers each year are divided into two volumes of three numbers each. Editorial correspondence Papers submitted for publication and editorial correspondence should be addressed to Maureen Schupsky, Annals of Mathematics, Fine Hall-Washington Road, Princeton University, Princeton, NJ, 08544-1000 U.S.A. The e-mail address is [email protected]. 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Canonical subgroups of Barsotti-Tate groups...... 955–988 Akshay Venkatesh. Sparse equidistribution problems, period bounds and subconvexity ...... 989–1094 Thomas Geisser. Duality via cycle complexes ...... 1095–1126 Viktor L. Ginzburg. The Conley conjecture ...... 1127–1180 Christopher Voll. Functional equations for zeta functions of groups and rings...... 1181–1218 mathematics of annals Monika Ludwig and Matthias Reitzner. A classification of SL.n/ invariant valuations ...... 1219–1267 Isaac Goldbring. Hilbert’s fifth problem for local groups...... 1269–1314 Robert M. Guralnick and Michael E. Zieve. Polynomials with PSL.2/ monodromy...... 1315–1359 Robert M. Guralnick, Joel Rosenberg and Michael E. Zieve.A new family of exceptional polynomials in characteristic two ...... 1361–1390 Raman Parimala and V. Suresh. The u-invariant of the function fields of p-adic curves ...... 1391–1405 Avraham Aizenbud, Dmitry Gourevitch, Stephen Rallis and Gerard´ Schiffmann. Multiplicity one theorems ...... 1407–1434 Stanislav Smirnov. Conformal invariance in random cluster models. I. Holmorphic fermions in the Ising model ...... 1435–1467 Kannan Soundararajan. Weak subconvexity for central values of L-functions ...... 1469–1498 Roman Holowinsky. Sieving for mass equidistribution ...... 1499–1516 Roman Holowinsky and Kannan Soundararajan. Mass equidistribution for Hecke eigenforms ...... 1517–1528 Kannan Soundararajan. Quantum unique ergodicity for SL2.ޚ/ ވ ...... 1529–1538 2010 September, n