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L$-functions with the Lindel\"of and Riemann hypotheses and trivial zeros at even negative integers

##article.authors##

  • Takashi Nakamura Institute of Arts and Sciences, Tokyo University of Science

DOI:

https://doi.org/10.51094/jxiv.620

Keywords:

L-functions, Lindeloef hypothesis, Riemann's functional equation, Riemann hypothesis, trivial zeros

Abstract

Let $q \ge 2$ be an integer, $\zeta (s)$ be the Riemann zeta function, and put $T_q (s) := (s+1)(1-s)^{-1} (q^{s+2}-1) \zeta (s+2) - 4 \pi^2 s^{-1} (1-s)^{-1}(q^{3-s}-1) \zeta (s-2)$. In the present paper, we show that the function $T_q (s)$ has Riemann's functional equation and its zeros only at the negative even integers and satisfies the Lindel\"of and Riemann hypotheses. In addition, we give functions satisfy Riemann's functional equation and an analogue of the Lindel\"of hypothesis but do not fulfill an analogue of the Riemann hypothesis.

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Submitted: 2024-02-13 06:05:39 UTC

Published: 2024-02-16 00:50:19 UTC — Updated on 2024-05-27 10:06:28 UTC

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Mathematics