• Bernoulli
  • Volume 7, Number 5 (2001), 817-828.

The Riemann zeta distribution

Gwo Dong Lin and Chin-Yuan Hu

Full-text: Open access


Let $\zeta$ be the Riemann zeta function. Khinchine (1938) proved that the function $f_\sigma(t)=\zeta(\sigma + $i $t)/\zeta(\sigma)$, where $\sigma > 1$ and $t$ is real, is an infinitely divisible characteristic function. We investigate further the fundamental properties of the corresponding distribution of $f_\sigma$, the Riemann zeta distribution, including its support and unimodality. In particular, the Riemann zeta random variable is represented as a linear function of infinitely many independent geometric random variables. To extend Khinchine's result, we construct the Dirichlet-type characteristic functions of discrete distributions and provide a sufficient condition for the infinite divisibility of these characteristic functions. By way of applications, we give probabilistic proofs for some identities in number theory, including a new identity for the reciprocal of the Riemann zeta function.

Article information

Bernoulli Volume 7, Number 5 (2001), 817-828.

First available in Project Euclid: 15 March 2004

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

completely multiplicative function Dirichlet series geometric distribution infinite divisibility Jordan totient function Liouville function Mangoldt function Möbius function Poisson distribution Riemann zeta distribution Riemann zeta function


Dong Lin, Gwo; Hu, Chin-Yuan. The Riemann zeta distribution. Bernoulli 7 (2001), no. 5, 817--828.

Export citation


  • [1] Apostol, T.M. (1976) Introduction to Analytic Number Theory. New York: Springer-Verlag.
  • [2] Chung, K.L. (2001) A Course in Probability Theory, 3rd edition. New York: Academic Press.
  • [3] Gnedenko, B.V. and Kolmogorov, A.N. (1968) Limit Distributions for Sums of Independent Random Variables, transl. K.L. Chung. Reading, MA: Addison-Wesley.
  • [4] Khinchine, A.Ya. (1938) Limit Theorems for Sums of Independent Random Variables (in Russian). Moscow and Leningrad: GONTI.
  • [5] Loève, M. (1977) Probability Theory I, 4th edition. New York: Springer-Verlag.
  • [6] Lukacs, E. (1970) Characteristic Functions, 2nd edition. New York: Hafner.
  • [7] Patterson, S.J. (1995) An Introduction to the Theory of the Riemann Zeta-Function. Cambridge: Cambridge University Press.