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# Extension of the Riemann $\xi$-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip

Published:2009-06-01
Printed: Jun 2009
• Kevin A. Broughan
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## Abstract

If $K$ is a number field with $n_k=[k:\mathbb{Q}]$, and $\xi_k$ the symmetrized Dedekind zeta function of the field, the inequality $$\Re\,{\frac{ \xi_k'(\sigma + {\rm i} t)}{\xi_k(\sigma + {\rm i} t)}} > \frac{ \xi_k'(\sigma)}{\xi_k(\sigma)}$$ for $t\neq 0$ 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 $\si\ge 1$ for all $t$ sufficiently large or small and to the region $\si\ge 1+1/(\log t -5)$ for all $t\neq 0$. This answers positively a question posed by Lagarias.
 Keywords: Riemann zeta function, xi function, zeta zeros
 MSC Classifications: 11M26 - Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses 11R42 - Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]