The Titchmarsh problem with integers having a given number of prime divisors

The asymptotics for the number of representations ofN asN→∞ is expressed as the sum of a number havingk prime divisors and a product of two natural numbers. The asymptotics is found fork≤(2−ε) ln lnN and (2+ε) ln lnN≤k≤b ln lnN, whereε>0. The results obtained are uniform with respect tok. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 585–602, April, 1996. This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00260.