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Article

Garunkštis, Ramūnas ; Grigutis, Andrius
The size of the Lerch zeta-function at places symmetric with respect to the line $\Re (s)=1/2$. (English). Czechoslovak Mathematical Journal, vol. 69 (2019), issue 1, pp. 25-37
MSC: 11M35 | MR 3923571 | Zbl 07088766 | DOI: 10.21136/CMJ.2018.0149-17
Keywords:
Lerch zeta-function; functional equation; zero distribution
Summary:
Let $\zeta (s)$ be the Riemann zeta-function. If $t\geq 6.8$ 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 $|L(\lambda ,\lambda ,1-\overline {s})|>|L(\lambda ,\lambda ,s)|$.
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