Sums involving the number of distinct prime factors function

2016/04/16 by Tanay Wakhare, Wakhare, Tanay
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #History and Overview (math.HO)

paper · pdf · doi:10.48550/arxiv.1604.05671

openalex publication_date 2016/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

The main object of this paper is to find closed form expressions for finite and infinite sums that are weighted by ω(n), where ω(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.

Citations

• Survey of Dirichlet Series of Multiplicative Arithmetic Functions

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