Please use this identifier to cite or link to this item: http://hdl.handle.net/10077/31872
Title: Some inequalities for the Riemann zeta function
Authors: Alzer, Horst
Kwong, Man Kam
Keywords: Riemann zeta functionEuler's constantvon Mangoldt functioninequalities
Issue Date: 2021
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
Journal: Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics 
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.
Type: Article
URI: http://hdl.handle.net/10077/31872
ISSN: 0049-4704
eISSN: 2464-8728
DOI: 10.13137/2464-8728/31872
Rights: Attribution-NonCommercial-NoDerivatives 4.0 Internazionale
Appears in Collections:Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.53 (2021)

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