On the mean square gap between primes

Abstract

We prove that the average size of the squares of differences between consecutive primes less than $x$ is $O(x^{0.23+\varepsilon})$ for any fixed $\varepsilon>0$. This improves on a result of Peck, who gave bound $O(x^{0.25+\varepsilon})$ in the place of $O(x^{0.23+\varepsilon})$. Key ingredients are Harman's sieve, Heath-Brown's mean value theorem for sparse Dirichlet polynomials and Heath-Brown's $R^*$ bound.