On the number of representations in the Waring-Goldbach problem with a prime variable in an arithmetic progression

Abstract

We prove a Bombieri-Vinogradov type theorem for the number of representations of an integer N in the form N=p 1 g +p 2 g +...+p s g with p 1 ,p 2 ,...,p s prime numbers such that p 1 ≡l( mod k), under suitable hypothesis on s=s(g) for every integer g≥2.