Distribution of numbers with a given number of prime divisors in progressions

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] …”

“…The following lemma, which we can be regarded as an analog of the Vinogradov-Bombieri theorem, was proved in [8].…”

“…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.…”

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

“…The first two lemmas are analogs of the VinogradovBobieri theorem and the Bruno--Titchmarsh inequality, respectively. They were proved in [9] (see Theorenan 1 and 2 and also the remark to Theorem 2). Denote ~(x,k,t,a,d)= l{n : n < x, a(n)=k, (n,P(t)) = l, n-a(modd)}l,…”

“…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.…”