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

The Riemann zeta distribution

Gwo Dong Lin and Chin-Yuan Hu

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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.

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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.

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