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An estimate of a Dirichlet series of Kloosterman type. (English) Zbl 0880.11043
Let \(\Gamma\) be a finitely generated Fuchsian group of the first kind with the cusp \(\infty\), and let \(q>0\) be minimal such that \(\left(\begin{smallmatrix} 1 & q\\ 0 & 1 \end{smallmatrix} \right)\) generates \(\Gamma_\infty\). For \(m\in\mathbb{Z}\), \(m\neq 0\) the author defines the exponential sum \[ S(m,c, \Gamma): =\sum_{0\leq a<qc} \exp \left(2 \pi i{ma \over qc} \right), \quad \left(\begin{matrix} a & * \\ c & d \end{matrix} \right) \in \Gamma\quad (c>0) \] and the Dirichlet series \[ \Phi_m (s,\Gamma): =\sum_{c>0} S(m,c, \Gamma)c^{-2s}. \] This series comes up in the \(m\)-th Fourier coefficient of the Fourier expansion of the Eisenstein series for \(\Gamma\) at the cusp \(\infty\), and this series also comes up in the zeroth Fourier coefficient of the nonholomorphic Poincaré series \(P_m(z,s, \Gamma)\).
The aim of the paper under review is to show that \(\Phi_m\) admits a meromorphic continuation to the region \(\text{Re} s> {1\over 2}\) and to prove a growth estimate for this function in the domain \({1\over 2} <\text{Re} s <M\), \(|\text{Im} s |\geq 1\). To this end the author computes the inner product of \(P_m(z,s,\Gamma)\) with a certain series \(E_b(z,s, \Gamma)\) which is closely related with the Eisenstein series. Since \(\Phi_m\) comes up in the constant term of \(P_m\), this inner product can be expressed in terms of \(\Phi_m\).
11F37 Forms of half-integer weight; nonholomorphic modular forms
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F30 Fourier coefficients of automorphic forms
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