An average Brun-Titchmarsh theorem and shifted primes with a large prime factor

TL;DR

This paper refines the Brun-Titchmarsh theorem for large moduli using recent convolution estimates and applies these results to show infinitely many primes with large prime factors in their shifted form.

Contribution

It introduces an average Brun-Titchmarsh theorem for large moduli leveraging Maynard's recent estimates, improving previous bounds on shifted primes with large prime factors.

Findings

Largest prime factor of p-1 exceeds p^{0.679} infinitely often

Refined bounds on primes with large prime factors in shifted form

Enhanced understanding of prime distribution related to large prime factors

Abstract

The author studies an average version of Brun-Titchmarsh theorem with large moduli. Using Maynard's recent breakthrough on the Bombieri-Friedlander-Iwaniec type triple convolution estimates, we refine the previous result of Baker and Harman (1996). As an application, we improve a result of Baker and Harman (1998) on shifted primes with a large prime factor, showing that the largest prime factor of $p - 1$ is larger than $p^{0.679}$ for infinitely many primes $p$.