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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A modified Riemann zeta distribution in the critical strip
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by Takashi Nakamura PDF
Proc. Amer. Math. Soc. 143 (2015), 897-905 Request permission

Abstract:

Let $\sigma , t \in {\mathbb {R}}$, $s=\sigma +\textrm {{i}}t$ and $\zeta (s)$ be the Riemann zeta function. Put $f_\sigma (t):=\zeta (\sigma -\textrm {{i}}t)/(\sigma -\textrm {{i}}t)$ and $F_\sigma (t):= f_\sigma (t)/f_\sigma (0)$. We show that $F_\sigma (t)$ is a characteristic function of a probability measure for any $0 < \sigma \ne 1$ by giving the probability density function. By using this fact, we show that for any $C \in {\mathbb {C}}$ satisfying $|C| > 10$ and $-19/2 \le \Re C \le 17/2$, the function $\zeta (s) + Cs$ does not vanish in the half-plane $\sigma >1/18$. Moreover, we prove that $F_\sigma (t)$ is an infinitely divisible characteristic function for any $\sigma >1$. Furthermore, we show that the Riemann hypothesis is true if each $F_\sigma (t)$ is an infinitely divisible characteristic function for each $1/2 < \sigma <1$.
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Additional Information
  • Takashi Nakamura
  • Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo 3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan
  • MR Author ID: 755913
  • Email: takashin@ms.u-tokyo.ac.jp
  • Received by editor(s): April 9, 2013
  • Received by editor(s) in revised form: May 24, 2013, and June 3, 2013
  • Published electronically: October 31, 2014
  • Additional Notes: The author would like to thank the referees for their constructive and helpful comments and suggestions on the manuscript.
  • Communicated by: Mark M. Meerschaert
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 897-905
  • MSC (2010): Primary 60E10, 11M06; Secondary 60E07, 11M26
  • DOI: https://doi.org/10.1090/S0002-9939-2014-12279-9
  • MathSciNet review: 3283676