Hyperbolic summation involving the function Ω(n) and lcm

2022/12/11 by Meselem Karras, Karras, Meselem
Mathematics · #11A05 #11A25 #11N37 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2212.05440

openalex publication_date 2022/12/11 · openalex created_date 2022/12/26 · openalex updated_date 2026/07/28

Abstract

We study the sum ∑_{abc ≤ x} Ω([a,b,c]), where Ω(n) denotes the number of distinct prime divisors of n ∈ ℤ_{≥ 1}, counted with multiplicity, and where (a,b,c) = gcd(a,b,c) and [a,b,c] = lcm(a,b,c). An asymptotic formula is derived for this sum over the hyperbolic region \(a,b,c) ∈ ℤ_{≥ 1}³ : abc ≤ x\.

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