Zeta distribution

The zeta distribution is any 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

P(X=x) = x^-s/ζ(s)    for x = 1, 2, 3, ...

where s > 1 is a parameter and ζ(s) is Riemann's 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.