Geodesic
Functional Equations of Dirichlet Series Derived from Non-Analytic Automorphic Forms of a Certain Type
Canadian mathematical bulletin, Tome 18 (1975) no. 1, pp. 87-94

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 a positive real number and f(—1/τ) = γ(—iτ)kf(τ), k being a complex number. The function (—iτ)k is defined as exp(k log(—iτ) where log(—iτ) has the real value when —iτ is positive. Every such function is said to have signature (λ, k, γ) in the sense of E. Hecke [1] and has a Fourier expansion of the type f(τ) = a0 + σ an exp(2πin/λ), (n = 1,2,...), if we further assume that f(τ) = O(|y|-c) as y tends to zero uniformly for all x, c being a positive number. 
Rao, V. Venugopal. Functional Equations of Dirichlet Series Derived from Non-Analytic Automorphic Forms of a Certain Type. Canadian mathematical bulletin, Tome 18 (1975) no. 1, pp. 87-94. doi: 10.4153/CMB-1975-016-9
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[1] 1. Hecke, E., Über die Bestimmung Dirichletscher Reihen durch ihre Funchonalgleichung, Math. Annalen 112 (1936), 664-699. Google Scholar

[2] 2. Maass, H., Die Differentialgleichungen in der Théorie der elliptischen Modulfunktionen, Math. Annalen, 125 (1953), 235-263. Google Scholar

[3] 3. Maass, H., Über die räumliche Verteilung der Punk te in Git tern mit indéfini ter Metrik, Math. Annalen, 138 (1959), 287-315. Google Scholar

[4] 4. Magnus, W., Oberhettinger, F., and Soni, R. P., Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, New York, 1966. Google Scholar

[5] 5. Rao, V. Y., Averages involving Fourier Coefficients of Non-Analytic Automorphic Forms, Canadian Mathematical Bulletin, 13(2) (1970), 187-198. Google Scholar

[6] 6. Siegel, C. L., Über die Zetafunktionen indefiniter quadratischer Formen II, Math. Zeit, 44 (1939), 398-426. Google Scholar

[7] 7. Siegel, C. L., Indefinite Quadratischer Formen and Funktionentheorie I, Math. Annalen, 124 (1951), 17-54. Google Scholar

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