Inequalities for Riemann’s zeta function

Alzer, Horst

doi:10.4171/PM/1846

The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Portugaliae Mathematica

Full-Text PDF (57 KB) | Metadata |

Table of Contents |

PM summary

Online access to the full text of Portugaliae Mathematica is restricted to the subscribers of the journal, who are encouraged to communicate their IP-address(es) to their agent or directly to the publisher at
subscriptions@ems-ph.org

Volume 66, Issue 3, 2009, pp. 321–327

DOI: 10.4171/PM/1846

Published online: 2009-09-30

Inequalities for Riemann’s zeta function

Horst Alzer^[1]

(1) Waldbröl, Germany

Let ζ and Λ the the Riemann zeta function and the von Mangoldt function, respectively. Further, let c > 0. We prove that the double-inequality

exp(− c ∑^∞_{n = 1} ^Λ(n)/_n^s + α) < ^{ζ(s + c)}/_ζ(s) < exp(− c ∑^∞_{n = 1} ^Λ(n)/_n^s + β)

holds for all s > 1 with the best possible constants

α = 0 and β =1/log 2 log(c log 2/1 − 2^−c).

This extends and refines a recent result of Cerone and Dragomir.

Keywords: Riemann zeta function, von Mangoldt function, Euler totient function, Dirichlet series, inequalities

Alzer Horst: Inequalities for Riemann’s zeta function. Port. Math. 66 (2009), 321-327. doi: 10.4171/PM/1846