Minimal theta functions

Source abstract

Let be a positive definite binary quadratic form with real coefficients and discriminant b 2 − 4 ac = −1. Among such forms, let . The Epstein zeta function of f is denned to be Rankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0, We prove a corresponding result for theta functions. For real α > 0, let This function satisfies the functional equation (This may be proved by using the formula (4) below, and then twice applying the identity (8).)