Divisor functions in the ring of Gaussian integers weighted by the generalized Kloosterman sum (Q1873212)

In the ring \(R\) of Gaussian integers let \(\tau_{a,b,c}(\omega)\) denote the number of factorizations of \(\omega \in R\) in the form \(\omega = \omega_1^a \omega_2^b \omega_3^c\), where \(a,b,c\) are natural numbers. The author constructs an asymptotic formula for the sum \[ \sum_{N (\omega) \leq x} \tau_{1,b,c}(\omega) K_{1,b,c}(\gamma; \omega,1,1), \] where \(\tau_{1,b,c}(\omega)\) is weighted by a generalized Klosterman function of type \[ K_{a,b,c}(\gamma; \alpha_1, \alpha_2, \alpha_3)= \sum_{\delta_1^a \delta_2^b \delta_3^c \equiv 1 \mod \gamma} e^{\pi i Sp \frac{\alpha_1 \delta_1 + \alpha_2 \delta_2 + \alpha_3 \delta_3}{\gamma}}. \]