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
= 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
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
+ 1) − log10
), 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.