Zeta distribution

zeta
Probability mass function Plot of the Zeta PMF on a log-log scale. (The function is only defined at positive integer values of k. The connecting lines do not indicate continuity.)
Cumulative distribution function
Parameters	${\displaystyle s\in (1,\infty )}$
Support	${\displaystyle k\in \{1,2,\ldots \}}$
PMF	${\displaystyle {\frac {1/k^{s}}{\zeta (s)}}}$
CDF	${\displaystyle {\frac {H_{k,s}}{\zeta (s)}}}$
Mean	${\displaystyle {\frac {\zeta (s-1)}{\zeta (s)}}~{\textrm {for}}~s>2}$
Mode	${\displaystyle 1\,}$
Variance	${\displaystyle {\frac {\zeta (s)\zeta (s-2)-\zeta (s-1)^{2}}{\zeta (s)^{2}}}~{\textrm {for}}~s>3}$
Entropy	${\displaystyle \sum _{k=1}^{\infty }{\frac {1/k^{s}}{\zeta (s)}}\log(k^{s}\zeta (s)).\,\!}$
MGF	does not exist
CF	${\displaystyle {\frac {\operatorname {Li} _{s}(e^{it})}{\zeta (s)}}}$
PGF	${\displaystyle {\frac {\operatorname {Li} _{s}(z)}{\zeta (s)}}}$

In probability theory and statistics, the zeta distribution is a discrete probability distribution. If X is a zeta-distributed random variable with parameter s, then the probability that X takes the positive integer value k is given by the probability mass function

${\displaystyle f_{s}(k)={\frac {k^{-s}}{\zeta (s)}}}$

where ζ(s) is the Riemann zeta function (which is undefined for s = 1).

The multiplicities of distinct prime factors of X are independent random variables.

The Riemann zeta function being the sum of all terms ${\displaystyle k^{-s}}$ for positive integer k, it appears thus as the normalization of the Zipf distribution. The terms "Zipf distribution" and "zeta distribution" are often used interchangeably. But while the Zeta distribution is a probability distribution by itself, it is not associated with Zipf's law with the same exponent.