New bounds and progress towards a conjecture on the summatory function of \((-2)^{\Omega(n)}\)

• Published in: arXiv.org
• Main Authors: Johnston, Daniel R, Leong, Nicol, Tudzi, Sebastian
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 08.09.2024
• ISSN: 2331-8422

In this article, we study the summatory function \begin{equation*} W(x)=\sum_{n\leq x}(-2)^{\Omega(n)}, \end{equation*} where \(\Omega(n)\) 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 \(x\geq 1\). This result provides new progress towards a conjecture of Sun, which asks whether \(|W(x)|<x\) for all \(x\geq 3078\). To obtain our results, we computed new explicit bounds on the Mertens function \(M(x)\). These may be of independent interest. Moreover, we obtain similar results and make further conjectures that pertain to the more general function \begin{equation*} W_a(x)=\sum_{n\leq x}(-a)^{\Omega(n)} \end{equation*} for any real \(a>0\).