Bernoulli

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

The Riemann zeta distribution

Gwo Dong Lin and Chin-Yuan Hu

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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.

Article information

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

Dates
First available in Project Euclid: 15 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1079399543

Mathematical Reviews number (MathSciNet)
MR2003k:11134

Zentralblatt MATH identifier
0996.60013

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

Citation

Dong Lin, Gwo; Hu, Chin-Yuan. The Riemann zeta distribution. Bernoulli 7 (2001), no. 5, 817--828.https://projecteuclid.org/euclid.bj/1079399543


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References

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