A divisor function of Wigert and higher degree forms
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
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2026
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| _version_ | 1866914520421629952 |
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| author | Banerjee, Debika Dixit, Atul Gupta, Rajat |
| author_facet | Banerjee, Debika Dixit, Atul Gupta, Rajat |
| contents | Let $k\in\mathbb{N}$. Wigert's divisor function $d^{\left(\frac{1}{k}\right)}(j)$ counts the number of representations of $j$ of the form $m^k+mn$ with $m\geq1 , n\geq0$. Let $\mathcal{F}_k(s)$ denote the Dirichlet series of $d^{\left(\frac{1}{k}\right)}(j)$. While $\mathcal{F}_2(s)$ is essentially a well-known special case of the Euler-Zagier double zeta function, and hence well-studied, very little is known about $\mathcal{F}_k(s)$ for $k>2$. We offer three new representations for $\mathcal{F}_k(s)$ for $k\geq2$, one of which is an analogue of the Chowla-Selberg formula as well as of a formula of Atkinson. The meromorphicity of $\mathcal{F}_k(s)$ is also discussed. The special value $\mathcal{F}_3\left(\frac{3}{2}\right)$ is expressed in terms of an infinite series of Bessel functions and a generalized divisor function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27698 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A divisor function of Wigert and higher degree forms Banerjee, Debika Dixit, Atul Gupta, Rajat Number Theory Classical Analysis and ODEs Primary 11A25, 11M41. Secondary 33C20, 30B40 Let $k\in\mathbb{N}$. Wigert's divisor function $d^{\left(\frac{1}{k}\right)}(j)$ counts the number of representations of $j$ of the form $m^k+mn$ with $m\geq1 , n\geq0$. Let $\mathcal{F}_k(s)$ denote the Dirichlet series of $d^{\left(\frac{1}{k}\right)}(j)$. While $\mathcal{F}_2(s)$ is essentially a well-known special case of the Euler-Zagier double zeta function, and hence well-studied, very little is known about $\mathcal{F}_k(s)$ for $k>2$. We offer three new representations for $\mathcal{F}_k(s)$ for $k\geq2$, one of which is an analogue of the Chowla-Selberg formula as well as of a formula of Atkinson. The meromorphicity of $\mathcal{F}_k(s)$ is also discussed. The special value $\mathcal{F}_3\left(\frac{3}{2}\right)$ is expressed in terms of an infinite series of Bessel functions and a generalized divisor function. |
| title | A divisor function of Wigert and higher degree forms |
| topic | Number Theory Classical Analysis and ODEs Primary 11A25, 11M41. Secondary 33C20, 30B40 |
| url | https://arxiv.org/abs/2604.27698 |