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 and P a,c,η := {p ∈ P : p = P + (q − a) for some prime q with p η < q c(η)p η }.In this paper, under the Elliott-Halberstam conjecture we prove, for y → ∞,according to values of η. These complement for some results of Banks-Shparlinski [1], of Wu [12] and of Chen-Wu [2].

حول وجود تخمين K-Tuples الرئيسي للنسبة الإيجابية من K-Tuples المقبولة

“…where max is taken over all "a" which is coprime to "q", then the Elliott-Halberstam conjecture 6 states that for every 𝜃 < 1 and A > 0, ∃ a constant C such that:…”