## A modified Riemann zeta distribution in the critical strip

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## 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 $1/2 < \sigma <1$.## References

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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