Additive problems for integers with a given number of prime divisors

Abstract

AnSTRXCT. The asymptotics of sums of the form ~ r (Ibn -a[) (summation over n < N, w(n) = k) is studied, where to(n) is the number of distinct prime divisors of n, and ~-(n) is the number of all divisors.KEY WORDS: prime divisor, Titehmarsh problem, asymptotics of the sum of divisors.Let w(n) be the number of distinct prime divisors of the integer n, and let r(n) be the number of divisors of n. The goal of the present paper is to find the asymptotics of the sum r(Ib -al), where n is equal to the product of two primes satisfying certain conditions. For example, in [6] it was shown that n has at most six prime divisors greater than N l/as3 .