Zeta distribution - Definition and Overview

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

<math>P(X=x)=x^{-s}/\zeta(s)<math>

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.

If A is any set of positive integers that has a density, i.e., if

<math>\lim_{n\rightarrow\infty}\frac{\mbox{number}\ \mbox{of}\ \mbox{members}\ \mbox{of}\ A}{n}<math>

exists, then

<math>\lim_{s\rightarrow 1+}P(X\in A)<math>

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 log₁₀(d + 1) − log₁₀(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.