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AN ASYMPTOTIC ESTIMATE FOR THE CHARACTERISTIC AND NUMBER OF FIXED POINTS OF THE RIEMANN ZETA FUNCTION

Part of: Entire and meromorphic functions, and related topics Zeta and $L$-functions: analytic theory Series expansions

Published online by Cambridge University Press:  01 August 2025

BAO QIN LI Open the ORCID record for BAO QIN LI [Opens in a new window] ,
JÖRN STEUDING  and
YUTA SUZUKI
BAO QIN LI*
Affiliation:
Department of Mathematics and Statistics https://ror.org/02gz6gg07 Florida International University Miami , Florida 33199 United States
JÖRN STEUDING
Affiliation:
Department of Mathematics https://ror.org/00fbnyb24 University of Würzburg 97074 Würzburg Germany steuding@mathematik.uni-wuerzburg.de
YUTA SUZUKI
Affiliation:
Department of Mathematics https://ror.org/00x194q47 Rikkyo University 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501 Japan suzuyu@rikkyo.ac.jp
*
libaoqin@fiu.edu
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Abstract

We will give a precise and explicit asymptotic estimate for the characteristic of the Riemann zeta function $\zeta $ with an error term of order $O(\frac {\log r}{r})$ and a corresponding asymptotic estimate for the number of fixed points of $\zeta $.

Keywords

the Riemann zeta functionmeromorphic functionfixed pointcounting functioncharacteristic function

MSC classification

Primary: 11M06: $zeta (s)$ and $L(s, chi)$ 30B50: Dirichlet series and other series expansions, exponential series
Secondary: 30D35: Distribution of values, Nevanlinna theory

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Type
Article
Information
Nagoya Mathematical Journal , First View , pp. 1 - 18
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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