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
f s ( k ) = k − s ζ ( s ) {\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 k − s {\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.
Definition The zeta distribution is defined for positive integers k ≥ 1 {\displaystyle k\geq 1} , and its probability mass function is given by
P ( x = k ) = 1 ζ ( s ) k − s , {\displaystyle P(x=k)={\frac {1}{\zeta (s)}}k^{-s},}
where s > 1 {\displaystyle s>1} is the parameter, and ζ ( s ) {\displaystyle \zeta (s)} is the Riemann zeta function. The cumulative distribution function is given by
P ( x ≤ k ) = H k , s ζ ( s ) , {\displaystyle P(x\leq k)={\frac {H_{k,s}}{\zeta (s)}},}
where H k , s {\displaystyle H_{k,s}} is the generalized harmonic number
H k , s = ∑ i = 1 k 1 i s . {\displaystyle H_{k,s}=\sum _{i=1}^{k}{\frac {1}{i^{s}}}.}
Moments The nth raw moment is defined as the expected value of Xn:
m n = E ( X n ) = 1 ζ ( s ) ∑ k = 1 ∞ 1 k s − n {\displaystyle m_{n}=E(X^{n})={\frac {1}{\zeta (s)}}\sum _{k=1}^{\infty }{\frac {1}{k^{s-n}}}}
The series on the right is just a series representation of the Riemann zeta function, but it only converges for values of s − n {\displaystyle s-n} that are greater than unity. Thus:
m n = { ζ ( s − n ) / ζ ( s ) for n < s − 1 ∞ for n ≥ s − 1 {\displaystyle m_{n}={\begin{cases}\zeta (s-n)/\zeta (s)&{\text{for }}n<s-1\\\infty &{\text{for }}n\geq s-1\end{cases}}}
The ratio of the zeta functions is well-defined, even for n > s − 1 because the series representation of the zeta function can be analytically continued. This does not change the fact that the moments are specified by the series itself, and are therefore undefined for large n.
Moment generating function The moment generating function is defined as
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