The zeta distribution in probability theory is any one of a certain parametrized family of discrete probability distributions, whose support is the set of positive integers.
It can be defined by saying that if X is a random variable with a zeta distribution, then
for x = 1, 2, 3, ..., where s > 1 is a parameter and ζ(s) is the Riemann zeta function.
It can be shown that these are the only probability distributions for which the multiplicities of distinct prime factors of X are independent random variables.
Some applied statisticians have used the zeta distribution to model various phenomena; see the article on Zipf's law.
The case s = 1
ζ(1) is infinite as the harmonic series, and so the case when s = 1 is not meaningful. However, if A is any set of positive integers that has a density, i.e. if
exists, then
is equal to that density.
The latter limit can also exist in some cases in which A does not have a density. For example, if A is the set of all positive integers whose first digit is d, then A has no density, but nonetheless the second limit given above exists and is equal to
- log10(d + 1) − log10(d),
in accord with Benford's law.
See also
