Hassen, Abdulkadir (Professor)Gebreegziabher, Tewelde2020-08-142023-11-042020-08-142023-11-042020-07-14http://etd.aau.edu.et/handle/123456789/22072Chanrasekharan 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. In [5] Hawkins and Knopp proved a Hecke correspondence theorem for modular integrals with rational period function on Γθ, a sub group of the full modular group Γ(1). Sister Ann M. Heath in [1] considered the functional equation in the Hawkins and Knopp context. Analogous to Chandrasekharan and Narasimhan she showed its equivalence to two arithmetical identities associated with entire modular cusp integrals involving rational period functions for the full modular group. In this dissertation we extend the results of Sister Ann M. Heath to entire automorphic integrals involving rational period functions on discrete Hecke group G(λ), λ > 0. Moreover, we establish equivalence of two arithmetical identities with a functional equation associated with automorphic integrals involving log-polynomial-period functions on the Hecke groups.enDirichlet Series With Functional EquationsAutomorphic IntegralsArithmetical IdentitiesThesis