Primes in short arithmetic progressions

TL;DR

This paper develops a large sieve inequality for functions supported on primes, applies it to prove Elliott's conjecture, and provides bounds for short character sums over primes using sieve techniques.

Contribution

Introduces a new large sieve inequality for prime-supported functions and applies it to resolve a conjecture and bound character sums.

Findings

Proved Elliott's conjecture.

Established bounds for short character sums over primes.

Developed a novel combination of large sieve and Selberg sieve techniques.

Abstract

We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes. The proves uses a combination of the large sieve and the Selberg sieve.