Inequalities for Riemann's zeta function
Theorem. Let \(c>0\) be a real number. For all \(s>1\) we have \[ \exp\Big(-c\sum_{n=1}^{\infty} {{\Lambda(n)}\over {n^{s+\alpha}}}\Big) < {\zeta(s+c)\over \zeta(s)} < \exp\Big(-c\sum_{n=1}^{\infty} {{\Lambda(n)}\over {n^{s+\beta}}}\Big) \leqno(1) \] with the best possible constants \[ \alpha = 0 ,\quad \beta ={1\over{\log 2}}\log{\Big({{c \log 2}\over {{1-2^{-c}}}}}\Big).\leqno(2) \] This improves the double inequality by \textit{P. Cerone} and \textit{S. S. Dragomir} [Math. Nachr. 282, No. 7, 964--975 (2009; Zbl 1181.26030)] who proved (1) with ``\(\leq\)'' instead of ``\(<\)'' and \(\alpha =0, \beta =c = 1\).
scientific article; zbMATH DE number 5598199
scientific article