Geodesic
Averages Involving Fourier Coefficients of Non-Analytic Automorphic Forms
Canadian mathematical bulletin, Tome 13 (1970) no. 2, pp. 187-198

Voir la notice de l'article provenant de la source Cambridge University Press

Let f(τ) be a complex valued function, defined and analytic in the upper half of the complex τ plane (τ=x+iy, y > 0), such that f(τ+λ) = f(τ) where λ is real and f(-1/τ) = γ(-iτ)k f(τ), k being a complex number. The function (—iτ)k is defined as e k log(-iτ) where log(—iτ) has the real value when — iτ is positive and γ is a complex number with absolute value 1. Such functions have been studied by E. Hecke [4] who calls them functions with signature (λ, k, γ). We further assume that f(τ) = O(|y| -c ) as y tends to zero uniformly for all x, c being a positive real number. It then follows that f(τ) has a Fourier expansion of the type f(τ) = a 0 + Σ a n exp(2πinτ/λ) (n = 1,2,...), the series being convergent absolutely in the upper half plane. 
Rao, V. Venugopal. Averages Involving Fourier Coefficients of Non-Analytic Automorphic Forms. Canadian mathematical bulletin, Tome 13 (1970) no. 2, pp. 187-198. doi: 10.4153/CMB-1970-039-6
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