Version 1
: Received: 31 December 2023 / Approved: 2 January 2024 / Online: 2 January 2024 (09:43:48 CET)
How to cite:
Serguei, S. On the Sums over Inverse Powers of Zeroes of the Hurwitz Zeta-Function, and Some Related Properties of These Zeroes. Preprints2024, 2024010068. https://doi.org/10.20944/preprints202401.0068.v1
Serguei, S. On the Sums over Inverse Powers of Zeroes of the Hurwitz Zeta-Function, and Some Related Properties of These Zeroes. Preprints 2024, 2024010068. https://doi.org/10.20944/preprints202401.0068.v1
Serguei, S. On the Sums over Inverse Powers of Zeroes of the Hurwitz Zeta-Function, and Some Related Properties of These Zeroes. Preprints2024, 2024010068. https://doi.org/10.20944/preprints202401.0068.v1
APA Style
Serguei, S. (2024). On the Sums over Inverse Powers of Zeroes of the Hurwitz Zeta-Function, and Some Related Properties of These Zeroes. Preprints. https://doi.org/10.20944/preprints202401.0068.v1
Chicago/Turabian Style
Serguei, S. 2024 "On the Sums over Inverse Powers of Zeroes of the Hurwitz Zeta-Function, and Some Related Properties of These Zeroes" Preprints. https://doi.org/10.20944/preprints202401.0068.v1
Abstract
Recently, we have applied the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to find the sums over inverse powers of zeroes for the incomplete Gamma- and Riemann zeta- functions, polygamma functions, and elliptical functions. Here, the same theorem is applied to study such sums for the zeroes of the Hurwitz zeta-function , including the sum over the inverse first power of its appropriately defined non-trivial zeroes. We also study some related properties of the Hurwitz zeta-function zeroes. In particular, we show that for any natural N and small real epsilon, when z tends to n=0, -1, -2… we can find at least N zeroes of zeta in the - vicinity of 0 for sufficiently small epsilon, as well as one simple zero tending to 1, etc.
Keywords
Logarithm of an analytical function; Generalized Littlewood theorem; Hurwitz zeta-function; zeroes and poles of analytical function
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
