# zbMATH — the first resource for mathematics

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 $$\Phi_m$$.
##### MSC:
 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
Full Text: