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A modified Riemann zeta distribution in the critical strip


Author: Takashi Nakamura
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
Published electronically: October 31, 2014
MathSciNet review: 3283676
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Abstract: Let $ \sigma , t \in {\mathbb{R}}$, $ s=\sigma +{\rm {i}}t$ and $ \zeta (s)$ be the Riemann zeta function. Put $ f_\sigma (t):=\zeta (\sigma -{\rm {i}}t)/(\sigma -{\rm {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 $ \vert C\vert > 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
Email: takashin@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-2014-12279-9
Keywords: Characteristic function, infinite divisibility, zeros of zeta functions
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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.