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


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


Published: October 2001
First available in Project Euclid: 15 March 2004

zbMATH: 0996.60013
MathSciNet: MR2003K:11134

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

Rights: Copyright © 2001 Bernoulli Society for Mathematical Statistics and Probability

Vol.7 • No. 5 • October 2001
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