Dirichlet Series With Functional Equations Automorphic Integrals And Arithmetical Identities

Chanrasekharan and Narasimhan in [2] have shown that the functional equation Γ(s)ϕ(s) = Γ(δ−s)ψ(δ−s) is equivalent to two arithmetical identities. Inrn[5] Hawkins and Knopp proved a Hecke correspondence theorem for modularrnintegrals with rational period function on Γθ, a sub group of the full modularrngroup Γ(1).rnSister Ann M. Heath in [1] considered the functional equation in the Hawkinsrnand Knopp context. Analogous to Chandrasekharan and Narasimhan shernshowed its equivalence to two arithmetical identities associated with entirernmodular cusp integrals involving rational period functions for the full modularrngroup.rnIn this dissertation we extend the results of Sister Ann M. Heath to entirernautomorphic integrals involving rational period functions on discrete Heckerngroup G(λ), λ > 0. Moreover, we establish equivalence of two arithmeticalrnidentities with a functional equation associated with automorphic integralsrninvolving log-polynomial-period functions on the Hecke groups.

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