Additive problems with some conditions (Q1897862)

In dealing with asymptotics for the sum \(\sum_{n \leq x} f(n) \cdot d_k (n) \cdot r(n - 1)\), where \(d_k (n)\) is the number of distinct representations of \(n\) by \(k\) divisors, \(r(n)\) is the number of representations of \(n\) as the sum of two squares, and \(|f(n) |\leq 1\), problems of probabilistic number theory are solved. In particular, necessary and sufficient conditions are found for the weak convergence of some sequences of distribution functions to a limit distribution function.