On integrals of fractional parts and the theory of prime differences

TL;DR

This paper explores integrals of fractional parts and introduces new methods to analyze prime differences, providing insights into longstanding conjectures like Cramér's conjecture on prime gaps.

Contribution

It develops novel tools for understanding prime differences and offers simplified approaches to complex conjectures in number theory.

Findings

Demonstrates how to approach prime gap conjectures under simple assumptions

Provides new analytical tools for fractional parts of functions

Offers partial results related to Cramér's conjecture

Abstract

In this paper we study the integrals of fractional parts of given functions, and develop some new tools to understand the behaviour of prime differences. We demonstrate how simply some seemingly difficult conjectures related to prime differences can be dealt with. Some, good results discussed here includes, the well know conjecture on prime gaps by Cram\'er and $\lim \inf_{n\to\infty} d_n < \infty$. Based on some simple assumptions, we have demonstrated how to tackle such problems.