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

Broughan, Kevin A.

Canadian mathematical bulletin, Tome 52 (2009) no. 2, pp. 186-194

Résumé

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.

Détail

DOI : 10.4153/CMB-2009-021-3

Mots-clés : 11M26, 11R42, Riemann zeta function, xi function, zeta zeros