Combinatorial Models of the Distribution of Prime Numbers

Abstract

This work is divided into two parts. In the first one, the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed, and new integral lower and upper bounds of π(x) are found.

Keywords

• set partitions
• stirling numbers of the second kind
• harmonic numbers
• prime number distribution
• <i>Riemann Hypothesis</i>
• Gumbel distribution