A Novel Generalization of the Liouville Function $λ(n)$ and a Convergence Result for the Associated Dirichlet Series
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910902045900800 |
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| author | Waterpeace, Sky Pelletier |
| author_facet | Waterpeace, Sky Pelletier |
| contents | We introduce a novel arithmetic function $w(n)$, a generalization of the Liouville function $λ(n)$, as the coefficients of a Dirichlet series. By spatially encoding information in a natural way about the distribution of prime factors among natural numbers, $w(n)$ allows results to be obtained which rely intrinsically on the distribution of primes without having direct knowledge of that distribution. We prove some properties of the distribution of $w(n)$ and then provide a result on the convergence of its Dirichlet series. A parametrized family of functions $w_m(n)$ is defined of which $w(n)$ is a special case. We show that each function $w_m(n)$ injectively maps $\mathbb{N}$ into a dense subset of the unit circle in $\mathbb{C}$ and that each $F_m(s) = \sum_n \frac{w_m(n)}{n^s}$ converges for all $s$ with $\Re(s)\in\left(\frac{1}{2},1\right)$. Finally, we show that the family of functions $w_m(n)$ converges to $λ(n)$ and that $F_m(s)$ converges uniformly in $m$ to $\sum_n \frac{λ(n)}{n^s}$, implying convergence of that series in the same region and thereby proving an interesting property about a closely related function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_06867 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Novel Generalization of the Liouville Function $λ(n)$ and a Convergence Result for the Associated Dirichlet Series Waterpeace, Sky Pelletier Number Theory 11M26, 11M06 We introduce a novel arithmetic function $w(n)$, a generalization of the Liouville function $λ(n)$, as the coefficients of a Dirichlet series. By spatially encoding information in a natural way about the distribution of prime factors among natural numbers, $w(n)$ allows results to be obtained which rely intrinsically on the distribution of primes without having direct knowledge of that distribution. We prove some properties of the distribution of $w(n)$ and then provide a result on the convergence of its Dirichlet series. A parametrized family of functions $w_m(n)$ is defined of which $w(n)$ is a special case. We show that each function $w_m(n)$ injectively maps $\mathbb{N}$ into a dense subset of the unit circle in $\mathbb{C}$ and that each $F_m(s) = \sum_n \frac{w_m(n)}{n^s}$ converges for all $s$ with $\Re(s)\in\left(\frac{1}{2},1\right)$. Finally, we show that the family of functions $w_m(n)$ converges to $λ(n)$ and that $F_m(s)$ converges uniformly in $m$ to $\sum_n \frac{λ(n)}{n^s}$, implying convergence of that series in the same region and thereby proving an interesting property about a closely related function. |
| title | A Novel Generalization of the Liouville Function $λ(n)$ and a Convergence Result for the Associated Dirichlet Series |
| topic | Number Theory 11M26, 11M06 |
| url | https://arxiv.org/abs/2306.06867 |