New bounds and progress towards a conjecture on the summatory function of (-2)^Ω(n)

2024/08/08 by Daniel R. Johnston, Johnston, Daniel R., Nicol Leong +3
Mathematics · #11M41 (Primary) 11Y70 (Secondary) #11N56 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical Inequalities and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2408.04143

openalex publication_date 2024/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we study the summatory function W(x)=∑_{n≤ x}(-2)^Ω(n), where Ω(n) counts the number of prime factors of n, with multiplicity. We prove W(x)=O(x), and in particular, that |W(x)|<2260x for all x≥ 1. This result provides new progress towards a conjecture of Sun, which asks whether |W(x)|0.

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