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ABSTRACT. The uymptotics for the number of representations of N as N ---* oo is expressed as the sum of a number having h prime divisors and a product of two natural numbers. The asymptotics is found for k _< (2 -e) In in N and where p is a prime and r(n) is the number of divisors of n. Therefore, the binary additive problemsare called Titchmarsh problem~.The first definitive solution of these problems was obtained in the sixties by Yu. V. Linnik [2] within the framework of the variance method, which he had developed. I-Ie found the asymptotics of the number of solutions to the first equation (1) n 0 and ln2 N = lnlnN. Note that research in this direction began in [9], where the asymptotics of the sum (2) was obtained for n having k prime divisors k < (2 -e)ln2 N and a = 1. Let ~(n) be the number of prime divisors of n, counting multiplicity. Denote N(x, = I{": "< x, = k} I,

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