- 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
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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