Inequalities for Riemann’s zeta function

Abstract

Let $\zeta$ and $\Lambda$ be the Riemann zeta function and the von Mangoldt function, respectively. Further, let $c>0$. We prove that the double-inequality

$\exp (-c\sum _{n=1}^{\infty }\frac{\Lambda (n)}{n^{s+\alpha }})<\frac{\zeta (s+c)}{\zeta (s)}<\exp (-c\sum _{n=1}^{\infty }\frac{\Lambda (n)}{n^{s+\beta }})$

holds for all $s>1$ with the best possible constants

$\alpha =0\text{and}\beta =\frac{1}{\log 2}\log \left(\frac{c\log 2}{1-2^{-c}}\right).$

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