We 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 to use the theory of symmetric functions to derive identities for the elementary symmetric functions, then apply these identities to arbitrary primes and values of multiplicative functions evaluated at primes. This allows us to reinterpret sums over symmetric polynomials as divisor sums and sums over the natural numbers.
"Sums Involving the Number of Distinct Prime Factors Function,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 19:
1, Article 8.
Available at: https://scholar.rose-hulman.edu/rhumj/vol19/iss1/8