Sums involving the number of distinct prime factors function

TL;DR

This paper develops closed-form expressions and convergence criteria for sums weighted by the number of distinct prime factors, using polynomial theorems and factorization identities to analyze divisor sums and multiplicative functions.

Contribution

It introduces a novel approach combining polynomial theorems with multiplicative function analysis to evaluate and understand sums involving the ω(n) function.

Findings

Derived closed-form expressions for sums involving ω(n)

Established convergence criteria for these sums

Reinterpreted sums over complex numbers as divisor sums

Abstract

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.