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

Garunkštis, Ramūnas   ; Grigutis, Andrius

Czechoslovak Mathematical Journal, Tome 69 (2019) no. 1, pp. 25-37

Abstract (VO)

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 $|L(\lambda ,\lambda ,1-\overline {s})|>|L(\lambda ,\lambda ,s)|$.

Détail

DOI : 10.21136/CMJ.2018.0149-17

Classification : 11M35
Keywords: Lerch zeta-function; functional equation; zero distribution

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