On the number of restricted prime factors of an integer (Q397003)

Let \(q\geq 2\) and \(S(n)\) denote the sum of digits in base \(q\). Let \(\widetilde\omega(n)\) denote the number of distinct prime factors \(p\) of \(n\) such that \(S(p)\equiv a\mod b\). Similarly, let \(\widetilde \Omega(n)\) denote the number of prime factors \(p\) of \(n\) counted with multiplicities with \(S(p)\equiv a\mod b\). The authors estimate the summatory functions of \(\widetilde\omega\) and \(\widetilde\Omega\).