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On the Value Distribution of Hurwitz Zeta-Functions at the Nontrivial Zeros of the Riemann Zeta-Function

Authors:

Institution:

1. Mathematisches Seminar, Johann Wolfgang Goethe-Universit?t Frankfurt, Robert-Mayer-Str. 10, D-60054, Frankfurt, Germany

Abstract:

We consider the value distribution of Hurwitz zeta-functions $\zeta (s,\alpha ): = \sum\nolimits_{n = 0}^\infty {\frac{1}{{(n + \alpha )^s }}}$ at the nontrivial zeros ϱ= β + iγ of the Riemann zeta-function ζ (s):= ζ (s, 1). Using the method of Conrey, Ghosh and Gonek we prove for fixed 0< α< 1 andH ≤T that

$\begin{gathered} \sum\limits_{T< \gamma \leqslant T + H} {\zeta (\varrho ,\alpha ) = - \left( {\Lambda \frac{1}{\alpha } + \sum\limits_{n = 1}^\infty {\frac{{\exp ( - 2\pi i\alpha n}}{n}} } \right)\frac{H}{{2\pi }}} \hfill \\ + O(H exp( - C(log T)^{1/3} ) + T^{1/2 + \varepsilon } ) \hfill \\ \end{gathered}$

with some absolute constantC > 0 (a similar result was first proved by Fujii 4] under assumption of the Riemann hypothesis). It follows that $\frac{{\zeta (s,\alpha )}}{{\zeta (s)}}$ is an entire function if and only if α = 1/2 or α = l. Further, we prove for α ≠ 1/2, 1 the existence of zeros ϱ = β +iγ withT < γ ≤T + T^3/4, 1/2 β ≤ 9/10+ ε and ζ(ϱ,α)≠0.

Keywords:

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