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On the distribution of zeros of the Hurwitz zeta-function

Authors: Ramunas Garunkstis and Jörn Steuding
Journal: Math. Comp. 76 (2007), 323-337
MSC (2000): Primary 11M35, 11M26
Published electronically: October 11, 2006
MathSciNet review: 2261024
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Abstract: Assuming the Riemann hypothesis, we prove asymptotics for the sum of values of the Hurwitz zeta-function $ \zeta(s, \alpha)$ taken at the nontrivial zeros of the Riemann zeta-function $ \zeta(s)=\zeta(s,1)$ when the parameter $ \alpha$ either tends to $ 1/2$ and $ 1$, respectively, or is fixed; the case $ \alpha=1/2$ is of special interest since $ \zeta(s,1/2)=(2^s-1)\zeta(s)$. If $ \alpha$ is fixed, we improve an older result of Fujii. Besides, we present several computer plots which reflect the dependence of zeros of $ \zeta(s, \alpha)$ on the parameter $ \alpha$. Inspired by these plots, we call a zero of $ \zeta(s,\alpha)$ stable if its trajectory starts and ends on the critical line as $ \alpha$ varies from $ 1$ to $ 1/2$, and we conjecture an asymptotic formula for these zeros.

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Additional Information

Ramunas Garunkstis
Affiliation: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225 Vilnius, Lithuania

Jörn Steuding
Affiliation: Institut für Mathematik, Würzburg University, Am Hubland, 97074 Würzburg, Germany

Received by editor(s): March 3, 2005
Received by editor(s) in revised form: October 4, 2005
Published electronically: October 11, 2006
Additional Notes: The first author is partially supported by a grant from the Lithuanian State Science and Studies Foundation and also by INTAS grant no. 03-51-5070.
Article copyright: © Copyright 2006 American Mathematical Society