Extensions of some formulae of A. Selberg

Abstract

This paper is concerned with estimating the number of positive integers up to some bound (which tends to infinity), such that they have a fixed number of prime divisors, and lie in a given arithmetic progression. We obtain estimates which are uniform in the number of prime divisors, and at the same time, in the modulus of the arithmetic progression. These estimates take the form of a fixed but arbitrary number of main terms, followed by an error term.