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Extension of the Riemann ξ-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip
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- Journal:
- Canadian Mathematical Bulletin / Volume 52 / Issue 2 / 01 June 2009
- Published online by Cambridge University Press:
- 20 November 2018, pp. 186-194
- Print publication:
- 01 June 2009
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- Article
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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.
