Sieve Method and Prime Gaps via Probabilistic Method

TL;DR

This paper introduces a probabilistic approach to the sieve method for studying prime gaps, offering an alternative to traditional analytic and algebraic techniques, and connects these ideas with recent breakthroughs by Zhang and Maynard.

Contribution

It presents a novel probabilistic framework for the sieve method, providing new insights into prime gaps and linking to recent significant results in the field.

Findings

Probabilistic sieve method effectively analyzes prime gaps

Connections established with Zhang and Maynard's work

Potential for new proofs of prime gap results

Abstract

Most prime gaps results have been proven using tools from analytic or algebraic number theory in the last few centuries. In this paper, we would like to present some probabilistic way of proving many essential results. A major component of the proof is a probabilistic approach to the sieve method. In addition, we discuss their connections with recent work by Zhang and Maynard on small and large gaps in prime numbers.