[Xmd5a]

The link is between numbers and words. Both can be decomposed into primitive components: primes and lyndon words (combinatorics). The Zipf distribution takes this shape when the vocabulary is infinite:

    P(r)=r^-s / Zeta(s)  with s > 1

Replace r^-s with (r + a)^-s, i.e. move onto Mandelbrot's rank-shifted model of Zipf's law and Riemann's Zeta function becomes Hurwitz Zeta function (ZetaHurwitz(s,a+1) with the ranks starting at 1).

Zipf law != Zipf distribution, still this is intriguing.

Integers and Prime Numbers: Deriving Zipf's Law (https://arxiv.org/abs/2403.12773)

> In a prime number decomposition of integers in a given set, the occurrence frequencies of prime numbers are shown to satisfy a general forms of Zipf's law.

The same holds for Lyndon words decomposition of random strings to a certain extent.