On the asymptotic behavior of some sums involving the number of prime divisors function

We consider sums of values of the composition of a real periodic arithmetic function and the number of prime divisors function over integers not exceeding a given bound. The prime divisors may be counted as with their multiplicity or without it, and we can restrict these divisors to the additional condition of belonging to some special set. This special set may be, for example, a sum of several arithmetic progressions with a given difference or imply an analog of prime number theorem with a power decrement in the remainder term. Moreover, instead of the number of prime divisors function we can consider an arbitrary real additive function that equals to one on primes. As an example of the periodic arithmetic function we can consider the Legendre symbol. In the paper we prove asymptotic formulae for such sums and investigate their behavior.
The proof uses the decomposition of the periodic arithmetic function into additive characters of the residue group, so the problem reduces to special trigonometric sums with the number of prime divisors function in the exponent. In order to establish asymptotic formulae for such sums we consider the corresponding Dirichlet series, accomplish its analytic continuation and make use of the Perron formula and complex integration method in specially adapted form.