3 ms·

What is the relationship between Riemann zeta function and Zipf's law? And between words and primes?

• Xmd5a 3mo ago
  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 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.