Riemann's zeta-function and the divisor problem. II
type: Item
instance of: scholarly article
title: Riemann's zeta-function and the divisor problem. II
zbMATH Open document ID: 0785.11045
DOI: 10.1007/BF02559498
author: Matti Jutila
published in: Arkiv för Matematik
publication date: 1994-04-26T00:00:00Z
review text: In the first paper of this series [Ark. Mat. 21, 75-96 (1983; Zbl 0513.10040)] the relationship between \[ E(T)= \int_ 0^ T |\zeta({\textstyle {1\over 2}}+it)|^ 2 dt- T(\log(T/2\pi)+2\gamma-1) \quad\text{and}\quad \Delta(x)= \sum_{n\leq x} d(n)-x(\log x+2\gamma-1) \] was explored. The present paper makes the connection more precise, by giving an approximate formula for a certain mean-value of \(E(T)\), in terms of \(\Delta(x)\). As a corollary it is shown that if \(\Delta(x)\ll_ \varepsilon x^{\beta+\varepsilon}\) for any \(\varepsilon>0\), then \(E(T)\ll_ \varepsilon T^{(1+2\beta)/5+\varepsilon}\). Thus the conjectured bound \(\Delta(x)\ll x^{1/4+\varepsilon}\) would imply that \(E(T)\ll T^{3/10+\varepsilon}\). This improves on the corresponding bound from the previous paper.
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 493365
zbMATH Keywords: Riemann Zeta-function
zbMATH Keywords: Dirichlet divisor problem
zbMATH Keywords: second moment
zbMATH Keywords: mean- value
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
cites work: THE MEAN VALUE OF THE ZETA-FUNCTION ON THE CRITICAL LINE
cites work: The mean-value of the Riemann zeta function
cites work: Q5820721
cites work: Exponential Sums with a Difference
cites work: Q3220660
cites work: On the divisor and circle problems
cites work: Riemann's zeta-function and the divisor problem
cites work: ON THE APPROXIMATE FUNCTIONAL EQUATION FOR ζ2(s) AND OTHER DIRICHLET SERIES
cites work: A generalization of Atkinson's formula to L-functions
description: scientific article; zbMATH DE number 493365