A joint limit theorem for the Riemann zeta-function in the space of analytic functions
type: Item
instance of: scholarly article
title: A joint limit theorem for the Riemann zeta-function in the space of analytic functions
zbMATH Open document ID: 1023.11041
DOI: 10.1023/A:1020230127481
author: R. Steuding
published in: Lithuanian Mathematical Journal
publication date: 2003-05-19T00:00:00Z
review text: Let \(H(D)\) denote the space of functions analytic in the critical strip \(D= \{ s\in \mathbb C: 1/2 < \Re s < 1 \}\) equipped with the topology of uniform convergence on compacta, and write the \(r\)-fold Cartesian product of \(H(D)\) by itself as \(H^{r}(D)\). By \(\mathcal B (S)\) denote the Borel sets of a space \(S\). Define the probability measure on \((H^{r}(D), \mathcal B (H^{r}(D)))\) as \[ P_{T}(A) = {1\over T}\operatorname {meas} \{\tau \in [0,T] ; (\zeta(s+ ik_{1}\tau),\ldots, \zeta(s+ ik_{r}\tau)) \in A \} , \] where the Lebesgue measure is meant and \(k_{j}\)'s are positive integers. In this paper it is proved that as \(T\to\infty\), \(P_{T}\) weakly converges to a probability measure \(P_{\zeta}\). The study is pursued in another paper [Lith. Math. J. 42, No. 4, 419-434 (2002; Zbl 1023.11042)] of the author where the explicit form of the limit measure \(P_{\zeta}\) is found.
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11K99
zbMATH DE Number: 1912624
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: probability measure
zbMATH Keywords: weak convergence
reviewed by: C. Y. Yildirim
MaRDI profile type: Publication
full work available at URL: a:1020230127481
OpenAlex ID: W2913588428
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description: scientific article; zbMATH DE number 1912624