On the distribution of numbers of a special form (Q1898276)

Let a set of arithmetic progressions \(\{kx+ \ell_i\}\) modulo \(k\) be given such that \((k, \ell_i)=1\), \(i=1, 2, \dots, r\), \(1\leq r\leq \varphi (k)\), where \(\varphi (k)\) is the Euler function. Let \(R\) denote the set of natural numbers whose canonical forms contain only prime numbers from this set. This paper considers the distribution of numbers of \(R\) in residue classes modulo \(q\).