LAUSR

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 → ∞ , π a , c , η ( x ) : = | ( 1 , x ] ∩ P a , c , η | ∼ π ( x ) or π a , c , η ( x ) ≫ a , η π ( x ) according to values of η. These are complement for some results of Banks-Shparlinski [1] , of Wu [12] and of Chen-Wu [2] .