L-functions with Riemann's functional equation and the Lindel\"of and Riemann hypotheses
DOI:
https://doi.org/10.51094/jxiv.620Keywords:
L-functions, Lindeloef hypothesis, Riemann's functional equation, Riemann hypothesis, real zerosAbstract
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
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Takashi Nakamura
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