On the order of growth of the Kloosterman zeta function (Q1187567)

Two estimates for the Kloosterman-Selberg zeta function \(Z_{m,n} (s)= \sum_c c^{-2s} S(m, n, c, \Gamma)\) for congruence subgroups \(\Gamma\) of the modular group are given: \[ Z_{m,n} (s)\ll_M q^{2\text{ Re } s}|\text mn|^{1/2} |{Im } s|^{1/2} (\text {Re } s- 1/2)^{-3} \] for \(\text{Re }s\in (1/2, M)\), \(|\text{Im } s|\geq 1\), with \(M>1\), and \(\tau\mapsto \tau +q\) a generator of the subgroup of \(\Gamma\) fixing the cusp \(\infty\). \[ Z_{m,n} (s) \ll q^{2\text{ Re } s} |mn |^{1/2} (\text{Re } s-1/ 2)^{-3} |s- 1/2 |^{-1} \] for \(1/2< \text{Re } s< 1/2+ \varepsilon\), \(|\text{Im } s|\leq 1\), with \(\varepsilon> 0\) chosen in such a way that the interval \((1/2, 1/2+ 2\varepsilon)\) does not contain singularities of \(Z_{m,n} (s)\). The factor \(|mn |^{1/2}\) is better than the factor \(|mn |\) in, e.g., Appendix E, \S 7, of \textit{D. A. Hejhal} [The Selberg trace formula for \(\text{PSL} (2, \mathbb{R})\); Lect. Notes Math. 1001, Springer Verlag (1983; Zbl 0543.10020)]. As a corollary, it is shown that \[ \sum_{c<x} c^{-1} S(m, n, c, \Gamma_0 (N)) \ll |mn |^{1/2} x^{1/6} (\log x)^2 \] for the Hecke congruence subgroups with \(N\leq 17\). The proof is based on an expression in earlier work of the author for the Fourier coefficients of Poincaré series of Selberg type [\textit{E. Yoshida}, Mem. Fac. Sci., Kyushu Univ., Ser. A 45, 1-17 (1991; Zbl 0741.11024)]. Quite long computations lead to an explicit formula for the scalar product of two Poincaré series, which is interesting in itself. One of the terms in this formula contains \(Z_{m,n} (s)\); the proof of the theorem consists of estimating the other terms. The author claims that all results generalize to Fuchsian groups of the first kind with cusps.