zeta distribution

The zeta distribution in probability theory is any one 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}/\zeta(s)\,

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

The case s = 1

ζ(1) is infinite as the harmonic series, and so the case when s = 1 is not meaningful. However, if A is any set of positive integers that has a density, i.e. if

\lim_{n\rightarrow\infty}\frac{\mbox{number of members of } A \mbox{ less than or equal to }n}{n}\,

exists, then

\lim_{s\rightarrow 1+}P(X\in A)\,

is equal to that density. The latter limit can also exist in some cases in which A does not have a density. For example, 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.

Zeta Distribution

The case s = 1

See also