Inequalities for Riemann’s zeta function

Abstract

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

exp (- c n = 1 \sum \infty \frac{Λ ( n )}{n ^{s + α}}) < \frac{ζ ( s + c )}{ζ ( s )} < exp (- c n = 1 \sum \infty \frac{Λ ( n )}{n ^{s + β}})

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

α = 0 and β = \frac{1}{lo g 2} lo g (\frac{c lo g 2}{1 - 2 ^{- c}}) .

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

Horst Alzer, Inequalities for Riemann’s zeta function. Port. Math. 66 (2009), no. 3, pp. 321–327