# 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

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.

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 still exists in some cases in which

*A* does not have a density. In particular, 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 log

_{10}(

*d* + 1) − log

_{10}(

*d*), in accord with

Benford's law.

Some applied statisticians have used the zeta distribution to model various phenomena; see the article on Zipf's law.