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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].

##### MSC:
 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