Czechoslovak Mathematical Journal, Vol. 69, No. 1, pp. 25-37, 2019


The size of the Lerch zeta-function at places symmetric with respect to the line $\Re(s)=1/2$

Ramunas Garunkštis, Andrius Grigutis

Received March 29, 2017.   Published online April 20, 2018.

Abstract:  Let $\zeta(s)$ be the Riemann zeta-function. If $t\geq6.8$ and $\sigma>1/2$, then it is known that the inequality $|\zeta(1-s)|>|\zeta(s)|$ is valid except at the zeros of $\zeta(s)$. Here we investigate the Lerch zeta-function $L(\lambda,\alpha,s)$ which usually has many zeros off the critical line and it is expected that these zeros are asymmetrically distributed with respect to the critical line. However, for equal parameters $\lambda=\alpha$ it is still possible to obtain a certain version of the inequality $|L(\lambda,\lambda,1-\overline{s})|>|L(\lambda,\lambda,s)|$.
Keywords:  Lerch zeta-function; functional equation; zero distribution
Classification MSC:  11M35
DOI:  10.21136/CMJ.2018.0149-17


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Affiliations:   Ramunas Garunkštis, Andrius Grigutis, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania, e-mail: ramunas.garunkstis@mif.vu.lt, andrius.grigutis@mif.vu.lt


 
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