Some inequalities for the Riemann zeta function

Abstract

Our main result states that for all real numbers s>1 we have

\gamma < s (\frac{\zeta'(s)}{\zeta(s)}+\frac{1}{s-1}). \eqno (\ast)

The constant lower bound \gamma is sharp. This refines an inequality published by Delange in 1987. Applications of (\ast) lead to a monotonicity theorem, namely, that

\frac{(s-1)\zeta(s)}{s^{\alpha}}

is strictly increasing on (1,\infty) if and only if \alpha \leq \gamma, and to additional inequalities for the zeta function.