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Portugaliae Mathematica

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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)/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 − 2c).

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