Explicit estimates for the Riemann zeta function
type: Item
instance of: scholarly article
title: Explicit estimates for the Riemann zeta function
zbMATH Open document ID: 1062.11056
author: Yuanyou F. Cheng
author: Sidney W. Graham
published in: Rocky Mountain Journal of Mathematics
publication date: 2005-03-10T00:00:00Z
review text: The authors apply van der Corput's method for the estimation of exponential sums to obtain useful explicit estimates for \(|\zeta(1/2+it)|\). Although not the strongest ones known, the bounds attained by the authors have the merit that they hold for all values of \(t\). For example, as a Corollary of their Theorem 3, it follows that \(|\zeta(1/2+it)| \leq 2.657\) if \(0 \leq t\leq \text{e}\), while for \(t\geq \text{e}\) one has \(|\zeta(1/2+it)| \leq 3t^{1/6}\log t\). The proofs rely on seven lemmas, in whose formulation and proofs care has been taken in order that all estimates are explicit. Applications of the results to prime numbers in short intervals are announced to be forthcoming.
reviewed by: Aleksandar Ivić
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11Y35
Mathematics Subject Classification ID: 11N05
zbMATH DE Number: 2143226
zbMATH Keywords: exponential sums
zbMATH Keywords: van der Corput's method
zbMATH Keywords: upper bound
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: explicit estimate
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069799
OpenAlex ID: W2013189454
Wikidata QID: Q105737525
cites work: Q5621389
cites work: An explicit upper bound for the Riemann zeta-function near the line \(\omega=1\)
cites work: Q4692762
cites work: Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients
cites work: Exponential Sums and the Riemann Zeta Function IV
cites work: Q3220660
cites work: Q4023619
cites work: Q4396948
cites work: Q4315110
cites work: Q5666618
cites work: Q5803461
DOI: 10.1216/RMJM/1181069799
description: scientific article; zbMATH DE number 2143226