Canadian Mathematical Society www.cms.math.ca
Search results

Search: MSC category 11R42 ( Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] )

 Expand all        Collapse all Results 1 - 1 of 1

1. CMB 2009 (vol 52 pp. 186)

Broughan, Kevin A.
 Extension of the Riemann $\xi$-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip 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 zerosCategories:11M26, 11R42

© Canadian Mathematical Society, 2013 : http://www.cms.math.ca/