Unified Theory of Zeta-Functions Allied to Epstein Zeta-Functions and Associated with Maass Forms

Abstract

In this paper, we shall establish a hierarchy of functional equations (as a G-function hierarchy) by unifying zeta-functions that satisfy the Hecke functional equation and those corresponding to Maass forms in the framework of the ramified functional equation with (essentially) two gamma factors through the Fourier–Whittaker expansion. This unifies the theory of Epstein zeta-functions and zeta-functions associated to Maass forms and in a sense gives a method of construction of Maass forms. In the long term, this is a remote consequence of generalizing to an arithmetic progression through perturbed Dirichlet series.

Keywords

• Fourier–Whittaker expansion
• Chowla–Selberg integral formula
• Maass forms
• Epstein zeta-function
• ramified functional equation