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Lower bounds for moments of Lfunctions

Edited by Peter Sarnak, New York University, New York, NY, and approved March 10, 2005 (received for review March 2, 2005)
Abstract
The moments of central values of families of Lfunctions have recently attracted much attention and, with the work of Keating and Snaith [(2000) Commun. Math. Phys. 214, 57–89 and 91–110], there are now precise conjectures for their limiting values. We develop a simple method to establish lower bounds of the conjectured order of magnitude for several such families of Lfunctions. As an example we work out the case of the family of all Dirichlet Lfunctions to a prime modulus.
A classical question in the theory of the Riemann zeta function asks for asymptotics of the moments , where k is a positive integer. A folklore conjecture states that the 2kth moment should be asymptotic to C_{k}T(log T) ^{k} ^{2} for a positive constant C_{k} . Only very recently with the work of Keating and Snaith (1) modeling the moments of ζ(s) by moments of characteristic polynomials of random matrices has a conjecture emerged for the value of C_{k} . This conjecture agrees with classical results of Hardy and Littlewood and Ingham (see ref. 2) in the cases k = 1 and k = 2, but for k ≥ 3 very little is known. Ramachandra (3) showed that for positive integers 2k, and HeathBrown (4) extended this for any positive rational number k. Titchmarsh (see theorem 7.19 of ref. 2) had previously obtained a smooth version of these lower bounds for positive integers k.
Analogously, given a family of Lfunctions an important problem is to understand the moments of the central values of these Lfunctions. Modeling the family of Lfunctions by using the choice of random matrix ensembles suggested by Katz and Sarnak (5) in their study of lowlying zeros, Keating and Snaith (6) have advanced conjectures for such moments. We illustrate these conjectures by considering three prototypical examples. The family of Dirichlet Lfunctions L(s, χ) as χ varies over primitive characters (mod q) is a unitary family, and it is conjectured that where k ∈ and C _{1,} _{k} is a specified positive constant. The family of quadratic Dirichlet Lfunctions L(s, χ _{d} ), where d is a fundamental discriminant and χ _{d} is the associated quadratic character, is a symplectic family and it is conjectured that where k ∈ and C _{2,} _{k} is a specified positive constant. The family of quadratic twists of a given newform f, L(s, f ⊗ χ _{d} ) (the Lfunction is normalized so that the central point is ); this is an orthogonal family and it is conjectured that where k ∈ , C _{3,} _{k} is a specified constant that depends on the form f.
While asymptotics in Eqs. 1–3 are known for small values of k, for large k these conjectures appear formidable. Further, the methods used to obtain lower bounds for moments of ζ(s) do not appear to generalize to this situation. In this article we describe a simple method that furnishes lower bounds of the conjectured order of magnitude for many families of Lfunctions, including the three prototypical examples given above. As a rough principle, it seems that whenever one can evaluate the first moment of a family of Lfunctions (with a little bit to spare) then one can obtain good lower bounds for all moments. We illustrate our method by giving a lower bound (of the conjectured order of magnitude) for the family of Dirichlet characters modulo a prime.
Theorem. Let k be a fixed natural number. Then for all large primes q
Our methods yield corresponding lower bounds for several other families, as well as other applications, for instance to fluctuations of matrix elements of Maass wave forms in the modular domain, which are not presented here. We remark also that we may take k to be any positive rational number ≥1 in the theorem. If k = r/s(≥1) is rational, then we achieve this by taking in the argument below.
Proof: Let x:= q ^{1/(2k)} be a small power of q, and set . We will evaluate and show that S _{2} ≪ q(log q) ^{k} ^{2} ≪ S _{1}. The theorem then follows from Hölder's inequality:
If then we may write where d _{ℓ}(n; x) denotes the number of ways of writing n as a _{1} ··· a _{ℓ} with each a_{j} ≤ x. As usual d _{ℓ}(n) will denote the ℓth divisor function, and note that d _{ℓ}(n; x) ≤ d _{ℓ}(n) with equality holding when n ≤ x.
We start with S _{2}. Note that and so Since the orthogonality relation for characters (mod q) gives that only the diagonal terms m = n survive. Thus Since d_{k} (n, x) ≤ d_{k} (n) and Σ _{n} _{≤} _{y} d_{k} (n)^{2}/n ∼ c_{k} (log y) ^{k} ^{2} for a positive constant c_{k} , we find that , as claimed.
We now turn to S _{1}. If Re(s) > 1 then integration by parts gives Since the numerator of the integrand above is log q by the PólyaVinogradov inequality (see chapter 23 of ref. 7) the above expression furnishes an analytic continuation of L(s, χ) to Re(s) > 0. Moreover we obtain We choose here X = q log^{4} q and obtain Since A(χ)^{2k–1} ≤ 1 + A(χ)^{2k} the error term above is ≪ (q + S _{2})/log q. The main term is Recalling that x = q ^{1/(2k)} and using the orthogonality relation for characters we conclude
The main term above will arise from the diagonal terms an = b. Let us first estimate the contribution of the offdiagonal terms. Here we may write an = b + qℓ, where 1 ≤ ℓ ≤ Xx^{k} ^{–1}/q = x^{k} ^{–1}(log q)^{4}. The contribution of these offdiagonal terms is since . Therefore, Since and d_{k} (b; x) = d_{k} (b) for b ≤ x, we deduce that This proves the theorem.
Acknowledgments
This research was partially supported by United StatesIsrael Binational Science Foundation Grant 2002088. K.S. was partially supported by the National Science Foundation.
Footnotes

↵ ‡ To whom correspondence should be addressed. Email: rudnick{at}post.tau.ac.il.

Author contributions: Z.R. and K.S. designed research, performed research, and wrote the paper.

This paper was submitted directly (Track II) to the PNAS office.
 Copyright © 2005, The National Academy of Sciences
References
 ↵

↵
Titchmarsh, E. C. (1986) The Theory of Riemann Zeta Function (Oxford Univ. Press, Oxford).

↵
Ramachandra, K. (1980) HardyRamanujan J. 3 , 1–25.

↵
HeathBrown, D. R. (1981) J. London Math. Soc. 24 , 65–78.

↵
Katz, N. & Sarnak, P. (1998) Random Matrices, Frobenius Eigenvalues, and Monodromy (Am. Soc. Math., Providence, RI).
 ↵

↵
Davenport, H. (1980) Multiplicative Number Theory (Springer, New York).
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