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

DOI:
https://doi.org/10.1090/S0025-5718-06-01882-5

Published electronically:
October 11, 2006

MathSciNet review:
2261024

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Abstract | References | Similar Articles | Additional Information

Abstract: Assuming the Riemann hypothesis, we prove asymptotics for the sum of values of the Hurwitz zeta-function taken at the nontrivial zeros of the Riemann zeta-function when the parameter either tends to and , respectively, or is fixed; the case is of special interest since . If is fixed, we improve an older result of Fujii. Besides, we present several computer plots which reflect the dependence of zeros of on the parameter . Inspired by these plots, we call a zero of *stable* if its trajectory starts and ends on the critical line as varies from to , 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

Email:
ramunas.garunkstis@maf.vu.lt

**Jörn Steuding**

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

Email:
steuding@mathematik.uni-wuerzburg.de

DOI:
https://doi.org/10.1090/S0025-5718-06-01882-5

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