A Vinogradov-type problem in almost primes

TL;DR

This paper generalizes Vinogradov's theorem to count sequences of almost primes with specified prime factor counts, providing asymptotic formulas for their distribution under certain conditions.

Contribution

It extends Vinogradov's theorem to almost primes with fixed prime factor counts, establishing asymptotics for their linear combinations.

Findings

Derived asymptotic formulas for sequences of almost primes.

Established conditions for the distribution of such sequences.

Extended classical results to a broader class of prime-related sequences.

Abstract

We prove a generalisation of Vinogradov's theorem by finding for $m\geqslant 3$ and fixed positive integers $c_1, \dots ,c_m, r_1, \dots , r_m$ the asymptotics of the number of sequences $(n_1, \dots ,n_m) \in \mathbf{N}^{m}$ such that $c_1n_1 + \dots + c_m n_m = N$ and $\Omega (n_i) = r_i$ for every $i=1, \dots ,m$ under the assumption that at least three of the $r_i$ are equal to $1$.