Markov processes on the adeles and Chebyshev function
type: Item
instance of: scholarly article
title: Markov processes on the adeles and Chebyshev function
zbMATH Open document ID: 1264.60059
author: Kumi Yasuda
published in: Statistics \& Probability Letters
publication date: 2013-01-25T00:00:00Z
review text: The author obtains a representation formula for the values of Chebyshev's function \(\psi(x)\) by the exit time \(T\) of adeles-valued processes from the finite integral adeles \(\mathbb{R}\times\prod_{p \in {\mathcal P}}\mathbb{Z}_p\), i.e., that there holds relation \(\psi(x)=\operatorname{E}(T(x))^{-1}\) (here \(\mathcal P\) is the set of all primes, \(\mathbb{R}\) is the set of all real numbers, \(\mathbb{Z}_p\) are the integer rings). From this relation, it is easy to see that the open assertion \(\zeta(s)\not =0\) if \(\operatorname{Re} s>\alpha\) for some \(\frac{1}{2} \leq \alpha <1\) can be described by the asymptotic behavior of the expectation of the exit times \(T(x)\) as \(x \to \infty\).
reviewed by: Roma Kačinskaitė
Mathematics Subject Classification ID: 60J27
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 6130789
zbMATH Keywords: Markov processes
zbMATH Keywords: Riemann zeta function
MaRDI profile type: Publication
full work available at URL: j.spl.2012.09.008
OpenAlex ID: W1965784971
DOI: 10.1016/J.SPL.2012.09.008
description: scientific article; zbMATH DE number 6130789