Gaps between prime divisors and analogues in Diophantine geometry

TL;DR

This paper advances understanding of prime divisor gaps by establishing asymptotics for all moments and demonstrating Poisson distribution of prime gaps related to rational points on varieties.

Contribution

It generalizes Erdős's work by providing asymptotic formulas for all moments of prime divisor gaps and shows these gaps follow a Poisson distribution in a geometric context.

Findings

Asymptotics for all moments of prime divisor gaps are derived.

Gaps between primes with no rational points on a variety are Poisson distributed.

Extends classical results to a geometric and probabilistic framework.

Abstract

Erd\H{o}s considered the second moment of the gap-counting function of prime divisors in 1946 and proved an upper bound that is not of the right order of magnitude. We prove asymptotics for all moments. Furthermore, we prove a generalisation stating that the gaps between primes $p$ for which there is no $\mathbb{Q}_p$-point on a random variety are Poisson distributed.