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.