Theta Series with Hecke Character

Abstract

In 1920 Hecke (Math. Z. 6, 11–51, 1920, Math. Werke 14, 249–289) introduced a new kind of theta series. The corresponding Dirichlet series form a common generalization both of Dirichlet’s L-series and of Dedekind’s zeta functions. While Dirichlet’s L-series are defined by characters on the rational integers, Hecke’s L-functions involve characters on the integral ideals of algebraic number fields. The values of these characters at principal ideals depend on the values of the algebraic conjugates a _ν of a generating number a, and not just on the residue of a modulo a fixed period ideal. Therefore Hecke called his characters Grössencharaktere. We prefer to use the term Hecke character. We can find definitions and results on Hecke characters, Hecke theta series and Hecke L-functions in some textbooks; we mention (Miyake in Modular Forms, Springer, Berlin, 1989), pp. 90–95, 182–185, (Neukirch in Algebraische Zahlentheorie, Springer, Berlin, 1992. English Translation: Algebraic Number Theory, Springer, Berlin, 1999), pp. 491–514. Here we will reproduce relevant definitions and results, but we will not give proofs.