Kloosterman sums and a mean value for Dirichlet polynomials
type: Item
instance of: scholarly article
title: Kloosterman sums and a mean value for Dirichlet polynomials
zbMATH Open document ID: 0837.11050
author: Nigel Watt
published in: Journal of Number Theory
publication date: 1996-05-20T00:00:00Z
review text: The Lindelöf conjecture is equivalent to the estimate, for any \(\varepsilon > 0\), \[ \int^T_0 \bigl |\zeta (1/2 + it) \bigr |^{2k} dt \leq c (\varepsilon, k) T^{1 + \varepsilon}, \quad k = 1,2,3, \dots, \] where \(\zeta\) is the Riemann zeta-function. This estimate has only been proved for \(k = 1\) and \(k = 2\). Even the proof of the above estimate for the case \(k = 3\) (the sixth power moment) would be a major advance in the subject. To study this problem one approach is to consider for a Dirichlet polynomial \(P(s) = \sum^M_{m = 1} a_m m^{- s}\), \(a_m\) complex, the mean value \[ I(T,M) = {1 \over T} \int^T_0 \bigl |P(it) \bigr |^2 \bigl |\zeta (1/2) + it) \bigr |^4 dt, \] for \(T \geq 1\). The Lindelöf conjecture is equivalent to \[ I(T,M) \ll T^\varepsilon \sum^M_{m = 1} |a_m |^2 \tag{*} \] for all \(M \leq T\), and the sixth power moment estimate would follow from this estimate for \(M = T^{1/2}\). \textit{H. Iwaniec} [J. Lond. Math. Soc., II. Ser. 22, 39-45 (1980; Zbl 0439.10026)] was the first to consider \(I(T,M)\) and he proved (*) for \(M \leq T^{1/10}\). \textit{J.-M. Deshouillers} and \textit{H. Iwaniec} [Mathematika 29, 202-212 (1982; Zbl 0506.10032)] improved this result by showing (*) holds for \(M \leq T^{1/5}\). Further, if the Selberg eigenvalue conjecture is true their method allowed \(M \leq T^{1/4}\). The present paper refines one aspect of the work of Deshouillers and Iwaniec to show that in the most interesting cases one can obtain this last result unconditionally by proving that \[ I(T,M) \ll T^\varepsilon (1 + M^2 T^{- 1/2}) M \max_{m \leq M} |a_m |^2. \] This result implies (*) for \(M \ll T^{1/4}\) provided the \(a_n\) satisfy \(a_n \ll_\varepsilon n^\varepsilon\) and \(\sum^n_{m = 1} |a_m |\gg_\varepsilon n^{1 - \varepsilon}\). This is the case in the most interesting examples where \(a_n\) is \(n^{ix}\) or \(\Lambda (n)\) or \(\tau (n)\). The author mentions that the new result should have applications to questions involving primes…
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 792476
zbMATH Keywords: mean value
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: Dirichlet polynomial
zbMATH Keywords: weighted large sieve inequality
zbMATH Keywords: exceptional eigenvalues
zbMATH Keywords: non-Euclidean Laplacian
zbMATH Keywords: Hecke congruence subgroups
zbMATH Keywords: Kuznetsov summation formula
zbMATH Keywords: sums of Kloosterman sums
reviewed by: Daniel Alan Goldston
MaRDI profile type: Publication
full work available at URL: jnth.1995.1086
OpenAlex ID: W2081795278
DOI: 10.1006/JNTH.1995.1086
description: scientific article; zbMATH DE number 792476