A heuristic study of the distribution of primes in short and not-so-short intervals

TL;DR

This paper presents a numerical heuristic analysis of prime distributions in short intervals, supporting conjectures about their variance based on Cramér's model and applicable for specific interval lengths.

Contribution

It introduces a heuristic expression for prime distribution variance in short intervals, extending understanding under Cramér's model and supporting Montgomery and Soundararajan's conjecture.

Findings

Heuristic expression for prime variance in short intervals.

Support for Montgomery and Soundararajan conjecture.

Validation of Cramér's model in this context.

Abstract

A numerical study on the distributions of primes in short intervals of length $h$ over the natural numbers $N$ is presented. Based on Cram\'er's model in Number Theory, we obtain a heuristic expression applicable when $h \gg \log{N}$ but $h \ll N$, providing support to the Montgomery and Soundararajan conjecture on the variance of the prime distribution at this scale.