Real analytic Eisenstein series

Eisenstein series are a fundamental topic in the theory of modular forms, particularly in the context of complex analysis and number theory. While the classical Eisenstein series are defined using complex variables, the concept can also be extended to the realm of real analysis, leading to the notion of real analytic Eisenstein series. ### Definition The real analytic Eisenstein series can be thought of as functions that are defined on the upper half-plane of complex numbers and exhibit certain symmetries under modular transformations.