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Zipf (Zeta) Distribution

docs/tutorial:stats:discrete_zipf

A random variable has the zeta distribution (also called the zipf distribution) with parameter $α > 1$ if its probability mass function is given by

where

ζ (α) = n = 1 \sum \infty \frac{1}{n ^{α}}

is the Riemann zeta function. Other functions of this distribution are

where $ζ_{i} = ζ (α - i)$ and $Li_{n} (z)$ is the $n^{th}$ polylogarithm function of $z$ defined as

Li_{n} (z) \equiv k = 1 \sum \infty \frac{z ^{k}}{k ^{n}}

μ_{n}^{'} = M^{(n)} (t)_{t = 0} = \frac{Li _{α - n} ( e ^{t} )}{ζ ( a )}_{t = 0} = \frac{ζ ( α - n )}{ζ ( α )}

Implementation: scipy.stats.zipf