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Spectral mean-values of automorphic \(L\)-functions at special points. (English) Zbl 0866.11034
Berndt, Bruce C. (ed.) et al., Analytic number theory. Vol. 2. Proceedings of a conference in honor of Heini Halberstam, May 16-20, 1995, Urbana, IL, USA. Boston, MA: Birkhäuser. Prog. Math. 139, 621-632 (1996).
Let \(u_j\) form an orthonormal basis of Maass cusp forms for the modular group \(\text{SL} (2,\mathbb{Z})\) acting by fractional linear transformation on the Poincaré upper half plane. This means that if \(\Delta=y^2 ({{\partial^2}\over {\partial x^2}}+ {{\partial^2}\over {\partial y^2}})\), \(\Delta u_j=s_j(s_j-1)u_j\). Moreover, \(u_j(z)=u_j ({{az+b}\over {cz+d}})\) for \(a,b,c,d\in\mathbb{Z}\) with \(ad-bc=1\) and \(u_j(z)\) goes to 0 as \(z\) goes to \(i\infty\). Assume as well that the \(u_j\) are eigenforms of all the Hecke operators \(T_n\) corresponding to the eigenvalue \(\lambda_j(n)\) and that they are eigenfunctions of the operator which sends \(f(z)\) to \(f(-\overline{z})\). Form the \(L\)-function \(L_j(s)=\sum^\infty_{n=1} \lambda_j(n)n^{-s}\), for \(\text{Re }s>1\). This function has analytic continuation to an entire function with functional equations. The main result is that for any \(\varepsilon>0\) and \(T\geq 1\), \[ \sum_{T<t_j< T+1}|L_j(s_j)|^4\ll T^{1+\varepsilon}. \] Here \(s_j={1\over 2}+it_j\). The proof requires an estimate for bilinear forms in Kloosterman sums and Bessel functions.
For the entire collection see [Zbl 0841.00015].

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11M41 Other Dirichlet series and zeta functions
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11L05 Gauss and Kloosterman sums; generalizations