On large differences between consecutive primes

TL;DR

This paper investigates large gaps between consecutive primes within intervals, providing bounds on the sum of such gaps, and applies these findings to prime representation, binary digit analysis, and real approximation.

Contribution

It introduces improved bounds on sums of large prime gaps using Heath-Brown's work and Harman's sieve, extending previous results in the field.

Findings

Bounds on sums of large prime gaps in intervals

Enhanced understanding of prime distribution with large gaps

Applications to prime-representing functions and digit analysis

Abstract

We show that $$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{1/2}}} (p_{n+1} - p_n) \ll x^{0.57+\epsilon}$$ and $$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{0.45}}} (p_{n+1} - p_n) \ll x^{0.63+\epsilon},$$ 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.