We investigate and generalise some properties of a family of probability distributions
closely related to the Riemann zeta function. Random variables
that have the property that divisibility by a set of distinct primes occurs
as a set of independent events are characterised in terms of functions that
are well known in number theory. We refer to random variables with this
independence property as Khinchin random variables.
In characterising the collection of Khinchin random variables, we make a
connection between the probabilistic theory of discrete distributions and the
number-theoretic concept of Dirichlet series. We outline some interesting
correspondences between discrete probability distributions and arithmetic
functions. A subset of the Khinchin random variables have infinitely divisible
logarithms. We establish the necessity of a condition, already known to be
sufficient, that ensures infinite divisibility.
Some Khinchin random variables admit a multiplicative decomposition
into a product of random prime numbers. The number of terms in such
a product follows a Poisson distribution. We explore two instances of this
decomposition: one related to the zeta distribution, and the other related to
the so-called prime zeta function.
We use the zeta distribution to derive known results from number theory
via probabilistic methods, and provide a generalisation of the distribution
for other unique factorisation domains.