Voronoi summation formula for the generalized divisor function $σ_{z}^{(k)}(n)$
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2023
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| author | Dixit, Atul Maji, Bibekananda Vatwani, Akshaa |
| author_facet | Dixit, Atul Maji, Bibekananda Vatwani, Akshaa |
| contents | For a fixed $z\in\mathbb{C}$ and a fixed $k\in\mathbb{N}$, let $σ_{z}^{(k)}(n)$ denote the sum of $z$-th powers of those divisors $d$ of $n$ whose $k$-th powers also divide $n$. This arithmetic function is a simultaneous generalization of the well-known divisor function $σ_z(n)$ as well as the divisor function $d^{(k)}(n)$ first studied by Wigert. The Dirichlet series of $σ_{z}^{(k)}(n)$ does not fall under the purview of Chandrasekharan and Narasimhan's fundamental work on Hecke's functional equation with multiple gamma factors. Nevertheless, as we show here, an explicit and elegant Vorono\"{\dotlessi} summation formula exists for this function. As its corollaries, some transformations of Wigert are generalized. The kernel $H_{z}^{(k)}(x)$ of the associated integral transform is a new generalization of the Bessel kernel. Several properties of this kernel such as its differential equation, asymptotic behavior and its special values are derived. A crucial relation between $H_{z}^{(k)}(x)$ and an associated integral $K_{z}^{(k)}(x)$ is obtained, the proof of which is deep, and employs the theory of linear differential equations and the properties of Stirling numbers of the second kind and elementary symmetric polynomials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_09937 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Voronoi summation formula for the generalized divisor function $σ_{z}^{(k)}(n)$ Dixit, Atul Maji, Bibekananda Vatwani, Akshaa Number Theory Classical Analysis and ODEs Primary 11M06, Secondary 33E20, 33C10 For a fixed $z\in\mathbb{C}$ and a fixed $k\in\mathbb{N}$, let $σ_{z}^{(k)}(n)$ denote the sum of $z$-th powers of those divisors $d$ of $n$ whose $k$-th powers also divide $n$. This arithmetic function is a simultaneous generalization of the well-known divisor function $σ_z(n)$ as well as the divisor function $d^{(k)}(n)$ first studied by Wigert. The Dirichlet series of $σ_{z}^{(k)}(n)$ does not fall under the purview of Chandrasekharan and Narasimhan's fundamental work on Hecke's functional equation with multiple gamma factors. Nevertheless, as we show here, an explicit and elegant Vorono\"{\dotlessi} summation formula exists for this function. As its corollaries, some transformations of Wigert are generalized. The kernel $H_{z}^{(k)}(x)$ of the associated integral transform is a new generalization of the Bessel kernel. Several properties of this kernel such as its differential equation, asymptotic behavior and its special values are derived. A crucial relation between $H_{z}^{(k)}(x)$ and an associated integral $K_{z}^{(k)}(x)$ is obtained, the proof of which is deep, and employs the theory of linear differential equations and the properties of Stirling numbers of the second kind and elementary symmetric polynomials. |
| title | Voronoi summation formula for the generalized divisor function $σ_{z}^{(k)}(n)$ |
| topic | Number Theory Classical Analysis and ODEs Primary 11M06, Secondary 33E20, 33C10 |
| url | https://arxiv.org/abs/2303.09937 |