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

Abstract

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<N where in [5,8] n is equal to the product of two primes pl < N a and p2 < N 1-" , in [6] n is equal to the product of k primes with pi < N =~ , ax < a2 < --" < a~ < 1/7, and al -t-a2 ~--.. -{-ak = 1. Finally, in [7] n has at most six prime divisors greater than N 1/ssa . Our aim is to find the asymptotics of the number of solutions to the equation n+xy=N, where x, y are natural numbers, and n has exactly k prime divisors, counting multiplicity. In contrast to previous work, we assume that k can vary with N, and the results obtained are uniform with respect to k <_ (2 -e)ln2 N and (2 + e)ln2 N < k < bln2 N, where e > 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,

Additive problems for integers with a given number of prime divisors

“…For example, in [6] it was shown that n has at most six prime divisors greater than N l/as3 . The present paper continues the previous work (see [8] and [9] …”

“…where a = k~ ln2 N, Unlike [8] and [9], in the proof of this theorem it is necessary to overcome additional difficulties arising from the fact that the properties of the functions fl(n) and w(n) are not the same. Despite the similarity of these functions, fl(n) is totally additive, while w(n) is only additive.…”