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The main object of this paper is to find closed form expressions for finiteand infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is thenumber of distinct prime factors of $n$. We then derive general convergencecriteria for these series. The approach of this paper is use polynomialtheorems to prove several factorization identities for arbitrary complexnumbers, then apply these identities with arbitrary primes and values ofmultiplicative functions evaluated at primes. This allows us to reinterpretsums over arbitrary complex numbers as divisor sums and sums over the naturalnumbers.

Sums involving the number of distinct prime factors function