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

Ramūnas Garunkštis; Andrius Grigutis

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 1, page 25-37
  • ISSN: 0011-4642

Abstract

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Let ζ ( s ) be the Riemann zeta-function. If t ≥ 6 . 8 and σ > 1 / 2 , then it is known that the inequality | ζ ( 1 - s ) | > | ζ ( s ) | is valid except at the zeros of ζ ( s ) . Here we investigate the Lerch zeta-function L ( λ , α , 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 λ = α it is still possible to obtain a certain version of the inequality | L ( λ , λ , 1 - s ¯ ) | > | L ( λ , λ , s ) | .

How to cite

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Garunkštis, Ramūnas, and Grigutis, Andrius. "The size of the Lerch zeta-function at places symmetric with respect to the line $\Re (s)=1/2$." Czechoslovak Mathematical Journal 69.1 (2019): 25-37. <http://eudml.org/doc/294293>.

@article{Garunkštis2019,
abstract = {Let $\zeta (s)$ be the Riemann zeta-function. If $t\ge 6.8$ 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 $|L(\lambda ,\lambda ,1-\overline\{s\})|>|L(\lambda ,\lambda ,s)|$.},
author = {Garunkštis, Ramūnas, Grigutis, Andrius},
journal = {Czechoslovak Mathematical Journal},
keywords = {Lerch zeta-function; functional equation; zero distribution},
language = {eng},
number = {1},
pages = {25-37},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The size of the Lerch zeta-function at places symmetric with respect to the line $\Re (s)=1/2$},
url = {http://eudml.org/doc/294293},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Garunkštis, Ramūnas
AU - Grigutis, Andrius
TI - The size of the Lerch zeta-function at places symmetric with respect to the line $\Re (s)=1/2$
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 1
SP - 25
EP - 37
AB - Let $\zeta (s)$ be the Riemann zeta-function. If $t\ge 6.8$ 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 $|L(\lambda ,\lambda ,1-\overline{s})|>|L(\lambda ,\lambda ,s)|$.
LA - eng
KW - Lerch zeta-function; functional equation; zero distribution
UR - http://eudml.org/doc/294293
ER -

References

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