New bounds and progress towards a conjecture on the summatory function of $(-2)^{Ω(n)}$
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916385964163072 |
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| author | Johnston, Daniel R. Leong, Nicol Tudzi, Sebastian |
| author_facet | Johnston, Daniel R. Leong, Nicol Tudzi, Sebastian |
| contents | In this article, we study the summatory function \begin{equation*}
W(x)=\sum_{n\leq x}(-2)^{Ω(n)}, \end{equation*} 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 $x\geq 1$. 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)^{Ω(n)} \end{equation*} for any real $a>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04143 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New bounds and progress towards a conjecture on the summatory function of $(-2)^{Ω(n)}$ Johnston, Daniel R. Leong, Nicol Tudzi, Sebastian Number Theory 11N56, 11M41 (Primary) 11Y70 (Secondary) In this article, we study the summatory function \begin{equation*} W(x)=\sum_{n\leq x}(-2)^{Ω(n)}, \end{equation*} 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 $x\geq 1$. 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)^{Ω(n)} \end{equation*} for any real $a>0$. |
| title | New bounds and progress towards a conjecture on the summatory function of $(-2)^{Ω(n)}$ |
| topic | Number Theory 11N56, 11M41 (Primary) 11Y70 (Secondary) |
| url | https://arxiv.org/abs/2408.04143 |