Moments of the Riemann zeta function and Eisenstein series. I
type: Item
instance of: scholarly article
title: Moments of the Riemann zeta function and Eisenstein series. I
zbMATH Open document ID: 1079.11041
author: Daniel Bump
author: Jennifer Beineke
published in: Journal of Number Theory
publication date: 2004-08-06T00:00:00Z
review text: The starting point and motivation of this paper is a comparison between the formulae \[ \int_0^T | \zeta (\sigma +it)| ^2\, dt \sim \zeta (2\sigma )T + \frac{(2\pi )^{2\sigma -1}}{2-2\sigma }\zeta (2-2\sigma )T^{2-2\sigma } \] and \[ \int_0^1 E_{\sigma }(x+iy)\, dx = \zeta (2 \sigma )y^{\sigma } + \pi ^{2\sigma -1}\frac{\Gamma (1-\sigma)}{\Gamma (\sigma )}\zeta (2-2\sigma )y^{1-\sigma }, \] where \(0 < \sigma < 1\) with \(\sigma \neq 1/2\), and \(E_{\sigma }(z)\) is the classical Eisenstein series for \(\text{SL}(2, \mathbb Z)\). Interestingly, the same zeta-values occur on the right in both relations. The authors show that the same phenomenon extends to higher moments of the zeta-function and higher dimensional Eisenstein series if the conjectural formula for the moments of the zeta-function, due to Conrey, Farmer, Keating, Rubinstein, and Snaith, is assumed. For Part II see J. Number Theory 105, No. 1, 175--191 (2004; Zbl 1079.11042).
reviewed by: Matti Jutila
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11F12
zbMATH DE Number: 2082499
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: Eisenstein series
MaRDI profile type: Publication
full work available at URL: j.jnt.2003.09.007
OpenAlex ID: W4213073983
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DOI: 10.1016/J.JNT.2003.09.007
description: scientific article; zbMATH DE number 2082499