On the value distribution of the Riemann zeta-function
type: Item
instance of: scholarly article
title: On the value distribution of the Riemann zeta-function
zbMATH Open document ID: 1243.30060
DOI: 10.1134/S1064562411030033
author: R. N. Boyarinov
published in: Doklady Mathematics
publication date: 2012-01-17T00:00:00Z
review text: The author continues to investigate the properties of the Riemann zeta function \(\zeta(s)\) on the critical strip \(0\leq \mathrm{Re} s\leq 1\). It is known that the function \(S(t):=\frac{1}{\pi}\zeta(\frac 12+it)\) changes the sign infinitely often as \(t\to\infty\). In 1946, A. Selberg developed a method and applied it to obtain a series of new results on the value distribution of the Riemann zeta function. In particular, he established a lower bound for the number of points at which \(S(t)\) changes sign in an interval \((T, T+H]\) of length \(H\). In the paper under review, the author obtains refinements of the lower and upper bounds for \(M(T+H)-M(T)\), where \(M(T)\) denotes the number of sign changes of \(S(t)\) in the interval \((0,T]\).
reviewed by: George Csordas
Mathematics Subject Classification ID: 30D15
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 5998392
zbMATH Keywords: value distribution
zbMATH Keywords: Riemann zeta function
MaRDI profile type: Publication
cites work: On the rate of convergence of distributions of random variables
cites work: On fractional moments of random variables
cites work: The zeros of the Riemann zeta-function
cites work: On the Riemann zeta-function -- mean value theorems and the distribution of \(|S(T)|\)
description: scientific article; zbMATH DE number 5998392