  The main object of this paper is to find closed form expressions for finite
and infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is the
number of distinct prime factors of $n$. We then derive general convergence
criteria for these series. The approach of this paper is use polynomial
theorems to prove several factorization identities for arbitrary complex
numbers, then apply these identities with arbitrary primes and values of
multiplicative functions evaluated at primes. This allows us to reinterpret
sums over arbitrary complex numbers as divisor sums and sums over the natural
numbers.
