## Bernoulli

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

### The Riemann zeta distribution

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 15 March 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1079399543

**Mathematical Reviews number (MathSciNet)**

MR2003k:11134

**Zentralblatt MATH identifier**

0996.60013

**Keywords**

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

#### Citation

Dong Lin, Gwo; Hu, Chin-Yuan. The Riemann zeta distribution. Bernoulli 7 (2001), no. 5, 817--828. https://projecteuclid.org/euclid.bj/1079399543