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

Canadian mathematical bulletin

关键词: Riemann zeta function;    xi function;    zeta zeros;   

DOI  :  10.4153/CMB-2009-021-3

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

If $K$ is a number field with $n_k=[k:mathbb{Q}]$, and $xi_k$the symmetrizedDedekind zeta function of the field, the inequality$$Re,{frac{ xi_k'(sigma + {m i} t)}{xi_k(sigma+ {m i} t)}} > frac{ xi_k'(sigma)}{xi_k(sigma)}$$ for $teq 0$ isshownto be true for $sigmage 1+ 8/n_k^frac{1}{3}$ improving the result ofLagarias where the constant in the inequality was 9. In the case $k=mathbb{Q}$theinequality is extended to $sige 1$ for all $t$ sufficiently large or smalland to the region $sige 1+1/(log t -5)$ for all $teq 0$. Thisanswers positively a question posed by Lagarias.

【 授权许可】

Unknown