Divisor function of the Gaussian integers weighted by the Kloosterman sum

• Date: 2026
• Main Authors: Varbanets, Pavel, Varbanets, Sergey, Vorobyov, Yakov
• Format: Article
• Language: English
• Published: Lugansk National Taras Shevchenko University 2026
• Subjects: exponential sums; Kloosterman sums; asymptotic formulas; divisor function; 11L05; 11L07
• Online Access: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2436
• Journal Title: Algebra and Discrete Mathematics

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.