ISSN:
1436-5081
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 + μ221E;)−s andL − (f, s) = ∑ n 〈 0 a ∞,n |n + μ221E;|−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).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01303063
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