The mean square of Dirichlet series associated with automorphic forms

Abstract

Letf be a non-holomorphic automorphic form of real weight and eigenvalue λ=1/4−ρ ², ℜρ≥0, which is defined with respect to a Fuchsian group of the first kind. Assume that ∞ is a cusp of this group and denote bya _∞,n,a _∞,n ,n ∈ ℤ, the Fourier coefficients off at ∞. Following Hecke and Maas we prove that under suitable assumptions the associated Dirichlet seriesL ⁺ (f, s) = ∑ _{n > 0} a _∞,n (n + μ_221E;)^−s andL ⁻ (f, s) = ∑ _{n < 0} a _∞,n |n + μ_221E;|^−s have meromorphic continuation in the entire complex plane and statisfy a certain functional equation (μ_∞ denotes the cusp parameter of the cusp ∞). We are interested in mean square estimates of these functions. Iff is not a cusp form we prove

$$\int_0^T {|L^ \pm (f,\Re _\rho + it)|^2 dt = T(\log T)^a (B^ \pm + o(1)),}$$

wherea is either 1, 2 or 4, andB ^± is a constant. A similar result is true iff is a cusp form. In case of a congruence group the termo(1) can be replaced byO ((logT)⁻¹).