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Extension of the Riemann ξ-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip

Published online by Cambridge University Press:  20 November 2018

Kevin A. Broughan
Kevin A. Broughan*
Affiliation:
Department of Mathematics, University of Waikato, Hamilton, New Zealand e-mail: kab@waikato.ac.nz
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Abstract

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If $K$ is a number field with ${{n}_{k}}\,=\,\left[ k\,:\,\mathbb{Q} \right]$, and ${{\xi }_{k}}$ the symmetrized Dedekind zeta function of the field, the inequality

$$\Re \frac{\xi _{k}^{'}\left( \sigma \,+\,\text{i}t \right)}{{{\xi }_{k}}\left( \sigma \,+\,\text{i}t \right)}\,>\,\frac{\xi _{k}^{'}\left( \sigma \right)}{{{\xi }_{k}}\left( \sigma \right)}$$

for $t\,\ne \,0$ is shown to be true for $\sigma \,\ge \,1\,+\,8/n_{k}^{\frac{1}{3}}$ improving the result of Lagarias where the constant in the inequality was 9. In the case $k\,=\,\mathbb{Q}$ the inequality is extended to $\sigma \,\ge \,1$ for all $t$ sufficiently large or small and to the region $\sigma \,\ge \,1\,+\,1/\left( \log \,t\,-\,5 \right)$ for all $t\,\ne \,0$. This answers positively a question posed by Lagarias.

Keywords

11M2611R42Riemann zeta functionxi functionzeta zeros
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 2 , 01 June 2009 , pp. 186 - 194
Copyright
Copyright © Canadian Mathematical Society 2009

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