Abstract
The primary concern of this paper is to deal with Siegel zeros of Hecke-Landau zeta-functions in an algebraic number field of finite degree over the rationals. As in the rational case with DirichletL-functions, the location of such zeros is closely connected with lower bounds for the corresponding zeta-functions at the points=1. This will be the theme in the first part of the paper. In this second part we first derive a form of the Brun-Titchmarsh theorem in the setting of a number field which is appropriate in our context. Then we turn our attention to the fact that an improvement of the constant in this inequality would lead to the nonexistence of Siegel zeros. The procedure is based on a weighted algebraic form of Selberg's upper bound sieve.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Fogels, E.: On the zeros of Hecke'sL-functions I. Acta Arith.7, 87–106 (1962).
Fogels, E.: Über die Ausnahmenullstelle der HeckeschenL-Funktionen. Acta Arith.8, 307–309 (1963).
Hinz, J.G.: Über Nullstellen der Heckeschen Zetafunktionen in algebraischen Zahlkörpern. Acta Arith.31, 167–193 (1976).
Hinz, J. G.: Character sums in algebraic number fields. J. Number Theory13, 463–484 (1981).
Hinz, J. G.: Eine Anwendung der Selbergschen Siebmethode in algebraischen Zahlkörpern. Acta Arith.41, 223–254 (1982).
Landau, E.: Über Ideale und Primideale in Idealklassen. Math. Z.2, 52–154 (1918).
Landau, E.: Verallgemeinerung eines Pólyaschen Satzes auf algebraische Zahlkörper. Göttinger Nachrichten, 478–488 (1918).
Motohashi, Y.: A note on Siegel's zeros. Proc. Japan Acad.55, 190–192 (1979).
Pintz, J.: Elementary methods in the theory ofL-functions I. Acta Arith.31, 53–60 (1976).
Prachar, K.: Primzahlverteilung. Berlin: Springer 1957.
Rieger, G. J.: Verallgemeinerung der Siebmethode von A. Selberg auf algebraische Zahlkörper II. J. Reine Angew. Math.201, 157–171 (1959).
Schaal, W.: Obere und untere Abschätzungen in algebraischen Zahlkörpern mit Hilfe des linearen Selbergschen Siebes. Acta Arith.13, 267–313 (1968).
Schaal, W.: On the large sieve method in algebraic number fields. J. Number Theory2, 249–270 (1970).
Tatuzawa, T.: On the number of integral ideals whose norms belonging to some norm residue class modq. Sci. Pap. Coll. Gen. Educ., Univ. Tokyo27, 1–8 (1977).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hinz, J., Lodemann, M. On Siegel zeros of Hecke-Landau zeta-functions. Monatshefte für Mathematik 118, 231–248 (1994). https://doi.org/10.1007/BF01301691
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01301691