The Zipf distribution, which is essentially the same as Zipf's law, results from the analysis of the frequencies of different words in almost all languages. As such it is an empirical distribution based on the observation that the number of words appearing r times, nr, in long sections of text can be modeled by the expression:
where k is a constant and ρ is a parameter; r is essentially the rank of the ordered set of words, with r=1 being the most frequent etc. By taking logs of both sides of this expression we have:
i.e. a linear equation on a log-log scaled chart. In fact the distribution is simply a discrete power law, much as the Pareto distribution is a continuous power law function. For the expression to form a probability distribution the constant, k, must be chosen to ensure that the terms sum to 1, hence we require that:
where ζ() denotes the Riemann zeta function, hence the alternative name for this distribution.
Mathworld/Weisstein E W: Zipf Distribution: https://mathworld.wolfram.com/ZipfDistribution.html
Wikipedia: Zipf's Law: https://en.wikipedia.org/wiki/Zipf%27s_law