On the difference between consecutive primes

TL;DR

This paper demonstrates that the sum of squared differences between consecutive primes up to a large number x is bounded by x^{5/4+ε}, using mean-value estimates for Dirichlet polynomials, reaffirming earlier results.

Contribution

Reproduces and confirms Peck's earlier result on bounds for the sum of squared prime gaps using similar proof techniques.

Findings

Sum of squared prime gaps is bounded by x^{5/4+ε} for large x.

The proof relies on mean-value estimates for Dirichlet polynomials.

No new results; reproduces earlier work.

Abstract

Update: This work reproduces an earlier result of Peck, which the author was initially unaware of. The method of the proof is essentially the same as the original work of Peck. There are no new results. We show that the sum of squares of differences between consecutive primes $\sum_{p_n\le x}(p_{n+1}-p_n)^2$ is bounded by $x^{5/4+{\epsilon}}$ for $x$ sufficiently large and any fixed ${\epsilon}>0$. The proof relies on utilising various mean-value estimates for Dirichlet polynomials.