On large differences between consecutive primes

Abstract

We show that $$\sum _{\begin{array}{c}{p}_n\in [x,2x]\\ p_{n+1}-p_n\ge x^{1/2}\end{array}}(p_{n+1}-p_n)\ll x^{0.57+\epsilon }$$ and $$\sum _{\begin{array}{c}{p}_n\in [x,2x]\\ p_{n+1}-p_n\ge x^{0.45}\end{array}}(p_{n+1}-p_n)\ll x^{0.63+\epsilon },$$ where $p_n$ is the $n$th prime number. The proof combines Heath-Brown's recent work with Harman's sieve, improving and extending his results. We give applications of the results to prime-representing functions, binary digits of primes and approximation of reals by multiplicative functions.

57 pages, 3 figures. Two ancillary files (C++-programs)