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ZEROS OF THE ESTERMANN ZETA FUNCTION

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  01 March 2013

A. DUBICKAS ,
R. GARUNKŠTIS ,
J. STEUDING  and
R. STEUDING
A. DUBICKAS*
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania
R. GARUNKŠTIS
Affiliation:
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania email ramunas.garunkstis@mif.vu.lt
J. STEUDING
Affiliation:
Department of Mathematics, University of Würzburg, Am Hubland, Campus-Nord, Emil Fischer Str. 40, 97074 Würzburg, Germany email steuding@mathematik.uni-wuerzburg.de
R. STEUDING
Affiliation:
Department of Mathematics, University of Würzburg, Am Hubland, Campus-Nord, Emil Fischer Str. 40, 97074 Würzburg, Germany email rasa.steuding@mathematik.uni-wuerzburg.de
*
✉arturas.dubickas@mif.vu.lt
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Abstract

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In this paper we investigate the zeros of the Estermann zeta function $E(s; k/ \ell , \alpha )= { \mathop{\sum }\nolimits}_{n= 1}^{\infty } {\sigma }_{\alpha } (n) \exp (2\pi ink/ \ell ){n}^{- s} $ as a function of a complex variable $s$, where $k$ and $\ell $ are coprime integers and ${\sigma }_{\alpha } (n)= {\mathop{\sum }\nolimits}_{d\vert n} {d}^{\alpha } $ is the generalized divisor function with a fixed complex number $\alpha $. In particular, we study the question on how the zeros of $E(s; k/ \ell , \alpha )$ depend on the parameters $k/ \ell $ and $\alpha $. It turns out that for some zeros there is a continuous dependency whereas for other zeros there is not.

Keywords

Estermann zeta functiondistribution of zerostrajectories of zeros

MSC classification

Secondary: 11M41: Other Dirichlet series and zeta functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 94 , Issue 1 , February 2013 , pp. 38 - 49
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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