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Some inequalities for the Riemann zeta function
Alzer, Horst
Kwong, Man Kam
2021
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e-ISSN
2464-8728
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.
Publisher
EUT Edizioni Università di Trieste
Source
Horst Alzer, Man Kam Wong, "Some inequalities for the Riemann zeta function" in: "Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.53 (2021)", EUT Edizioni Università di Trieste, Trieste, 2021. pp. 11
Languages
en
Rights
Attribution-NonCommercial-NoDerivatives 4.0 Internazionale
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