- Volume 7, Number 5 (2001), 817-828.
The Riemann zeta distribution
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.
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.https://projecteuclid.org/euclid.bj/1079399543