Integers without large prime factors

Abstract

Let u > 3 and β > √ e/(√ e−1) be real numbers and let β 0= β−√ e/(√ e−1). Let a and q be relatively prime positive integers. Let Ψ a ( X, Y) denote the number of positive integers ⩽ X and ≡ a (mod q), whose largest prime factor is ⩽ Y. There exists a computable q 0( u, β) such that, for q > q 0, ψ a(q uβ,q β)>q uβ−1 exp {− 2uβ β−1)√e ( log u+6+t)} where T = max { log log u, 4 + log ( β β 0)} .