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

##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、 trivial 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.

利益相反に関する開示

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

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引用文献

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投稿日時: 2024-02-13 06:05:39 UTC

公開日時: 2024-02-16 00:50:19 UTC — 2024-05-27 10:06:28 UTCに更新

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The title is changed. Section 1.2 and Some references are added.
研究分野
数学