International Journal of Mathematics and Mathematical Sciences
Volume 8 (1985), Issue 2, Pages 283-302

Extensions of some formulae of A. Selberg

Claudia A. Spiro

Department of Mathematics, State University of New York, Buffalo 14214, New York, USA

Received 3 March 1984

Copyright © 1985 Claudia A. Spiro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.