Divisor function of the Gaussian integers weighted by the Kloosterman sum
We study the mean values of the divisor function \(\tau(\omega)\) over the ring of Gaussian integers \(G\) when weighted by Kloosterman sums. For \(\alpha,\beta,\gamma\in{G}\) with \(\gamma\neq0\), let \(K(\alpha,\beta;\gamma)=\sum\limits_{x\in{G}_\gamma^\ast}\exp\left(2\pi{i}\Re\left(\frac{\alpha{x...
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| Дата: | 2026 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
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Lugansk National Taras Shevchenko University
2026
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2436 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543034122960896 |
|---|---|
| author | Varbanets, Pavel Varbanets, Sergey Vorobyov, Yakov |
| author_facet | Varbanets, Pavel Varbanets, Sergey Vorobyov, Yakov |
| author_sort | Varbanets, Pavel |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2026-01-11T10:11:21Z |
| description | We study the mean values of the divisor function \(\tau(\omega)\) over the ring of Gaussian integers \(G\) when weighted by Kloosterman sums. For \(\alpha,\beta,\gamma\in{G}\) with \(\gamma\neq0\), let \(K(\alpha,\beta;\gamma)=\sum\limits_{x\in{G}_\gamma^\ast}\exp\left(2\pi{i}\Re\left(\frac{\alpha{x}+\beta{x^{-1}}}{\gamma}\right)\right).\) We obtain an asymptotic formula for \(\sum\limits_{N(\omega)\leq{X}}\tau(\omega)\cdot{K}(1,\alpha\omega;\gamma),\) uniformly in \(\alpha\) co-prime to \(\gamma\) and with explicit dependence on \(N(\gamma)\). Our approach combines a Selberg–Kuznetsov–type identity over \(G\) with bounds for \(K(\alpha,\beta;\gamma)\) in prime-power modulus, together with Dirichlet–series methods for twisted sums \(Z_m(s;\delta_1,\delta_2)=\sum\limits_{\omega\in{G}}\frac{e^{4mi\arg(\omega+\delta_1)}\cdot{e}^{2\pi{i}\cdot\Re(\delta_2\omega)}}{N(\omega+\delta_1)^s}.\) We prove a truncated functional equation for \(Z_m\), establish mean-square bounds on the critical line, and deduce the required cancellation in the Kloosterman–weighted average of \(\tau(\omega)\). As by-products we record a generalized Selberg–Kuznetsov identity in \(G\) and Weil–type bounds for \(K(\alpha,\beta;\mathfrak{p}^m)\). These results extend classical techniques for \(\mathbb{Z}\) to the Gaussian setting and may be of independent interest for additive problems in \(G\) involving divisor-type functions and exponential sums. |
| first_indexed | 2026-02-08T07:56:48Z |
| format | Article |
| id | admjournalluguniveduua-article-2436 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:56:48Z |
| publishDate | 2026 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-24362026-01-11T10:11:21Z Divisor function of the Gaussian integers weighted by the Kloosterman sum Varbanets, Pavel Varbanets, Sergey Vorobyov, Yakov exponential sums, Kloosterman sums, asymptotic formulas, divisor function 11L05, 11L07 We study the mean values of the divisor function \(\tau(\omega)\) over the ring of Gaussian integers \(G\) when weighted by Kloosterman sums. For \(\alpha,\beta,\gamma\in{G}\) with \(\gamma\neq0\), let \(K(\alpha,\beta;\gamma)=\sum\limits_{x\in{G}_\gamma^\ast}\exp\left(2\pi{i}\Re\left(\frac{\alpha{x}+\beta{x^{-1}}}{\gamma}\right)\right).\) We obtain an asymptotic formula for \(\sum\limits_{N(\omega)\leq{X}}\tau(\omega)\cdot{K}(1,\alpha\omega;\gamma),\) uniformly in \(\alpha\) co-prime to \(\gamma\) and with explicit dependence on \(N(\gamma)\). Our approach combines a Selberg–Kuznetsov–type identity over \(G\) with bounds for \(K(\alpha,\beta;\gamma)\) in prime-power modulus, together with Dirichlet–series methods for twisted sums \(Z_m(s;\delta_1,\delta_2)=\sum\limits_{\omega\in{G}}\frac{e^{4mi\arg(\omega+\delta_1)}\cdot{e}^{2\pi{i}\cdot\Re(\delta_2\omega)}}{N(\omega+\delta_1)^s}.\) We prove a truncated functional equation for \(Z_m\), establish mean-square bounds on the critical line, and deduce the required cancellation in the Kloosterman–weighted average of \(\tau(\omega)\). As by-products we record a generalized Selberg–Kuznetsov identity in \(G\) and Weil–type bounds for \(K(\alpha,\beta;\mathfrak{p}^m)\). These results extend classical techniques for \(\mathbb{Z}\) to the Gaussian setting and may be of independent interest for additive problems in \(G\) involving divisor-type functions and exponential sums. Lugansk National Taras Shevchenko University 2026-01-11 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2436 10.12958/adm2436 Algebra and Discrete Mathematics; Vol 40, No 2 (2025) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2436/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2436/1342 https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/2436/1365 Copyright (c) 2026 Algebra and Discrete Mathematics |
| spellingShingle | exponential sums Kloosterman sums asymptotic formulas divisor function 11L05 11L07 Varbanets, Pavel Varbanets, Sergey Vorobyov, Yakov Divisor function of the Gaussian integers weighted by the Kloosterman sum |
| title | Divisor function of the Gaussian integers weighted by the Kloosterman sum |
| title_full | Divisor function of the Gaussian integers weighted by the Kloosterman sum |
| title_fullStr | Divisor function of the Gaussian integers weighted by the Kloosterman sum |
| title_full_unstemmed | Divisor function of the Gaussian integers weighted by the Kloosterman sum |
| title_short | Divisor function of the Gaussian integers weighted by the Kloosterman sum |
| title_sort | divisor function of the gaussian integers weighted by the kloosterman sum |
| topic | exponential sums Kloosterman sums asymptotic formulas divisor function 11L05 11L07 |
| topic_facet | exponential sums Kloosterman sums asymptotic formulas divisor function 11L05 11L07 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2436 |
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