On integrals of fractional parts and the theory of prime differences

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.