Elliott-Halberstam conjecture and values taken by the largest prime factor of shifted primes

Abstract

Denote by P the set of all primes and by P+(n) the largest prime factor of integer n⩾1 with the convention P+(1)=1. For each η>1, let c=c(η)>1 be some constant depending on η andPa,c,η:={p∈P:p=P+(q−a)for some prime q withpη<q⩽c(η)pη}. In this paper, under the Elliott-Halberstam conjecture we prove, for y→∞,πa,c,η(x):=|(1,x]∩Pa,c,η|∼π(x)orπa,c,η(x)≫a,ηπ(x) according to values of η. These are complement for some results of Banks-Shparlinski [1], of Wu [12] and of Chen-Wu [2].