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arXiv · 2408.04143

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

Daniel R. Johnston·Nicol Leong·Sebastian Tudzi
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Abstract

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 $|W(x)| 0$.

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BibTeXRIS

Daniel R. Johnston, Nicol Leong, Sebastian Tudzi. 2024-08-08. New bounds and progress towards a conjecture on the summatory function of $(-2)^{\Omega(n)}$. https://arxiv.org/abs/2408.04143

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