Opposite-sign Kloosterman sum zeta function (Q2810732)

The paper is devoted to the study of an opposite-sign Kloosterman sum zeta function NEWLINE\[NEWLINE Z_{m,-n}(s):=(2 \pi \sqrt{mn})^{2s-1}\sum_{l=1}^{\infty}\frac{S(m,-n,l)}{l^{2s}}, NEWLINE\]NEWLINE for \(n,m \in \mathbb Z_+\), \(s\in\mathbb C\).NEWLINENEWLINEIt is proved that the function \(Z_{m,-n}(s)\) has a meromorphic continuation to the entire complex plane \(\mathbb C\). The spectral expansion of opposite-sign Kloosterman sum zeta function as the sum of discrete and continuous spectrum is given (here the expression for the discrete spectrum relates with the problem of poles!). Also, a bound for the sum of opposite-sign Kloosterman sums, which is uniform in all the parameters, is provided.