Small Gaps Between Primes I

TL;DR

This paper uses divisor sums and moment analysis to show that a positive proportion of consecutive primes are within a quarter of their average spacing, advancing understanding of prime distribution.

Contribution

It introduces a novel approach connecting divisor sums and classical moment problems to analyze prime gaps.

Findings

A positive proportion of consecutive primes are within 25% of the average gap.

New methods link prime tuple approximations to classical moment problems.

Results improve bounds on small prime gaps.

Abstract

We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that a positive proportion of consecutive primes are within a quarter of the average spacing between primes.