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.
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References
E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. Zweite Mitteilung, Math. Z. 6 (1920), 11–51. Math. Werke 14, 249–289.
T. Miyake, Modular Forms, Springer, Berlin, 1989.
J. Neukirch, Algebraische Zahlentheorie, Springer, Berlin, 1992. English translation: Algebraic Number Theory, Springer, Berlin, 1999.
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© 2011 Springer-Verlag Berlin Heidelberg
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Köhler, G. (2011). Theta Series with Hecke Character. In: Eta Products and Theta Series Identities. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16152-0_5
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DOI: https://doi.org/10.1007/978-3-642-16152-0_5
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