Functional Equations of Dirichlet Series Derived from Non-Analytic Automorphic Forms of a Certain Type

Abstract

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) k f(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) = a 0 + σ a n 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.

Locations

• Canadian Mathematical Bulletin - View - PDF