Inequalities for Riemann’s zeta function

Abstract

sup { vertical-align: 0.8ex; font-size:85%; } sub { vertical-align: -0.8ex; font-size:85%; } .ionss { line-height: 1.8; } .ionss sub {position: relative; top:2; left:-12; } .ionss sup {position: absolute; top:34; left:181 } 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)/ns + α) < ζ(s + c)/ζ(s) < exp(− c ∑∞ n = 1 Λ(n)/ns + 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.