Inequalities for Riemann's zeta function
type: Item
instance of: scholarly article
title: Inequalities for Riemann's zeta function
zbMATH Open document ID: 1201.11086
DOI: 10.4171/PM/1846
published in: Portugaliae Mathematica. Nova Série
publication date: 2009-08-25T00:00:00Z
review text: Theorem. Let \(c>0\) be a real number. For all \(s>1\) we have \[ \exp\Big(-c\sum_{n=1}^{\infty} {{\Lambda(n)}\over {n^{s+\alpha}}}\Big) < {\zeta(s+c)\over \zeta(s)} < \exp\Big(-c\sum_{n=1}^{\infty} {{\Lambda(n)}\over {n^{s+\beta}}}\Big) \leqno(1) \] with the best possible constants \[ \alpha = 0 ,\quad \beta ={1\over{\log 2}}\log{\Big({{c \log 2}\over {{1-2^{-c}}}}}\Big).\leqno(2) \] This improves the double inequality by \textit{P. Cerone} and \textit{S. S. Dragomir} [Math. Nachr. 282, No. 7, 964--975 (2009; Zbl 1181.26030)] who proved (1) with ``\(\leq\)'' instead of ``\(<\)'' and \(\alpha =0, \beta =c = 1\).
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 5598199
zbMATH Keywords: Riemann zeta function
zbMATH Keywords: von Mangoldt function
zbMATH Keywords: Euler totient function
zbMATH Keywords: Dirichlet series
zbMATH Keywords: inequalities
author: Horst Alzer
reviewed by: Wolfgang Haneke
MaRDI profile type: Publication
cites work: Remark on a double-inequality for the Riemann zeta function
cites work: Q4101884
cites work: Euler and the Zeta Function
cites work: Some convexity properties of Dirichlet series with positive terms
cites work: Q4074972
cites work: Q3220660
cites work: The Riemann zeta-function. Transl. from the Russian by Neal Koblitz
cites work: Q5803461
full work available at URL: https://doi.org/10.4171/pm/1846
OpenAlex ID: W2048290940
Recommended article: Q5087765
Recommended article: An inequality for the Riemann zeta function
Recommended article: Q5173371
Recommended article: Q3476894
Recommended article: Rational inequalities and zeros of the Riemann zeta-function
Recommended article: Inequalities for the Hurwitz zeta function
Recommended article: Q4802109
description: scientific article; zbMATH DE number 5598199