CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

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
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   HTML   LaTeX   MathJax   PDF   PostScript  

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 Riemann zeta function, xi function, zeta zeros
MSC Classifications: 11M26, 11R42 show english descriptions Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]
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]
 

© Canadian Mathematical Society, 2014 : http://www.cms.math.ca/