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.
Citation
Gwo Dong Lin. Chin-Yuan Hu. "The Riemann zeta distribution." Bernoulli 7 (5) 817 - 828, October 2001.
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