L-functions with Riemann's functional equation and the Lindel\"of and Riemann hypotheses

Nakamura, Takashi

doi:10.51094/jxiv.620

##article.authors##

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

DOI:

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

キーワード:

L-functions、 Lindeloef hypothesis、 Riemann's functional equation、 Riemann hypothesis、 real zeros

抄録

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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The authors have no conflicts of interest directly relevant to the content of this article.

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