A Strong Symmetry Property of Eisenstein Series

Eisenstein series play a critical role in number theory. For two hundred years they have been an essential tool in the analysis of automorphic L-functions and in studying properties of quadratic forms in one and several variables. The construction is clear and straightforward, while their properties are sometimes very surprising. The arithmetic of their Fourier coefficients, and their analytic properties are still not completely understood. There are many connections with the Riemann hypothesis and other famous unsolved problems in number theory. Eisenstein series are named after Ferdinand Gotthold Eisenstein (1823 1852). Let k be an even integer larger than 2 and let τ be in the upper complex half-space. One of the simplest Eisenstein series is defined by