## Bernoulli

• Bernoulli
• Volume 21, Number 1 (2015), 604-617.

### A complete Riemann zeta distribution and the Riemann hypothesis

Takashi Nakamura

#### Abstract

Let $\sigma,t\in\mathbb{R}$, $s=\sigma+\mathrm{{i}}t$, $\Gamma(s)$ be the Gamma function, $\zeta(s)$ be the Riemann zeta function and $\xi(s):=s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$ be the complete Riemann zeta function. We show that $\Xi_{\sigma}(t):=\xi(\sigma-\mathrm{{i}}t)/\xi(\sigma)$ is a characteristic function for any $\sigma\in\mathbb{R}$ by giving the probability density function. Next we prove that the Riemann hypothesis is true if and only if each $\Xi_{\sigma}(t)$ is a pretended-infinitely divisible characteristic function, which is defined in this paper, for each $1/2<\sigma<1$. Moreover, we show that $\Xi_{\sigma}(t)$ is a pretended-infinitely divisible characteristic function when $\sigma=1$. Finally we prove that the characteristic function $\Xi_{\sigma}(t)$ is not infinitely divisible but quasi-infinitely divisible for any $\sigma>1$.

#### Article information

Source
Bernoulli, Volume 21, Number 1 (2015), 604-617.

Dates
First available in Project Euclid: 17 March 2015

https://projecteuclid.org/euclid.bj/1426597083

Digital Object Identifier
doi:10.3150/13-BEJ581

Mathematical Reviews number (MathSciNet)
MR3322332

Zentralblatt MATH identifier
1328.60048

#### Citation

Nakamura, Takashi. A complete Riemann zeta distribution and the Riemann hypothesis. Bernoulli 21 (2015), no. 1, 604--617. doi:10.3150/13-BEJ581. https://projecteuclid.org/euclid.bj/1426597083

#### References

• [1] Aoyama, T. and Nakamura, T. (2012). Multidimensional polynomial Euler products and infinitely divisible distributions on ${\mathbb{R}}^{d}$. Preprint, available at http://arxiv.org/abs/1204.4041.
• [2] Biane, P., Pitman, J. and Yor, M. (2001). Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions. Bull. Amer. Math. Soc. (N.S.) 38 435–465 (electronic).
• [3] Gnedenko, B.V. and Kolmogorov, A.N. (1968). Limit Distributions for Sums of Independent Random Variables, Revised ed. Reading, MA–London–Don Mills, ON: Addison-Wesley. Translated from the Russian, annotated, and revised by K.L. Chung. With Appendices by J.L. Doob and P.L. Hsu.
• [4] Gut, A. (2006). Some remarks on the Riemann zeta distribution. Rev. Roumaine Math. Pures Appl. 51 205–217.
• [5] Khintchine, A.Ya. (1938). Limits theorem for sums of independent random variables (in Russian). Moscow and Leningrad.
• [6] Lagarias, J.C. and Rains, E. (2003). On a two-variable zeta function for number fields. Ann. Inst. Fourier (Grenoble) 53 1–68.
• [7] Lin, G.D. and Hu, C.-Y. (2001). The Riemann zeta distribution. Bernoulli 7 817–828.
• [8] Lindner, A. and Sato, K.-i. (2011). Properties of stationary distributions of a sequence of generalized Ornstein–Uhlenbeck processes. Math. Nachr. 284 2225–2248.
• [9] Nakamura, T. A modified Riemann zeta distribution in the critical strip. Proc. Amer. Math. Soc. To appear.
• [10] Niedbalska-Rajba, T. (1981). On decomposability semigroups on the real line. Colloq. Math. 44 347–358.
• [11] Nikeghbali, A. and Yor, M. (2009). The Barnes $G$ function and its relations with sums and products of generalized gamma convolution variables. Electron. Commun. Probab. 14 396–411.
• [12] Patterson, S.J. (1988). An Introduction to the Theory of the Riemann Zeta-Function. Cambridge Studies in Advanced Mathematics 14. Cambridge: Cambridge Univ. Press.
• [13] Sato, K. Stochastic integrals with respect to Levy processes and infinitely divisible distributions. Sugaku Expositions. To appear.
• [14] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press. Translated from the 1990 Japanese original, Revised by the author.
• [15] Titchmarsh, E.C. (1986). The Theory of the Riemann Zeta-Function, 2nd ed. New York: Oxford Univ. Press. Edited and with a preface by D.R. Heath-Brown.
• [16] Whittaker, E.T. and Watson, G.N. (1996). A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions, Reprint of the fourth (1927) ed. Cambridge Mathematical Library. Cambridge: Cambridge Univ. Press.