Divisor function of the Gaussian integers weighted by the Kloosterman sum

Abstract

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.