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A zero-free region for the Hecke L-functions

Published online by Cambridge University Press:  26 February 2010

M. D. Coleman
Affiliation:
Department of Mathematics, University of Manchester Institute of Science and Technology, P.O. Box 88, Manchester, M60 1QD
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Extract

Let K be an algebraic number field of degree n and discriminant d. Let K(1),…, K(n) be the embeddings of the field. Then n = r1 + 2r2 where K(1), …, K(rl) are real and the remainder complex, satisfying . The conjugates of the number μ in K(i) are denoted by μ(i.

Type
Research Article
Copyright
Copyright © University College London 1990

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References

1. Borevich, Z. I. and Shafarevich, I. R.. Number Theory (Academic Press, 1966).Google Scholar
2. Coleman, M. D.. The distribution of points at which binary quadratic forms are prime. To appear in Proc. London Math. Soc.Google Scholar
3. Fogels, E.. On the zeros of Hecke's L-functions I. Acta Arith., 7 (1962), 87106.CrossRefGoogle Scholar
4. Hecke, E.. Eine neue Art von Zeta Funktionen und ihre Beziehung zur Verteilung der Primzahlen. I; II. Math. Z., 1 (1918), 357376; 6 (1920), 11–51.Google Scholar
5. Hecke, E.. Vorlesungen über die Theorie der algebraischen Zahlen, 2nd ed. (Chelsea, New York, 1970).Google Scholar
6. Hinz, J. G.. Eine Erweiterung des nullstellenfreien Bereiches der Heckeschen Zeta Funktion und Primideale in Idealklassen. Acta Arith., 38 (1980), 209253.CrossRefGoogle Scholar
7. Ivic, A.. The Riemann Zeta-Function (Wiley-Interscience, New York, 1985).Google Scholar
8. Kubilius, J. P.. On some problems of the geometry of prime numbers. Mat. Sb. N.S., 31 (1952), 507542.Google Scholar
9. Mitsui, T.. On the prime ideal theorem. J. Math. Soc. Japan, 20 (1968), 233247.CrossRefGoogle Scholar
10. Rademacher, H.. On the Phragmen-Lindelof Theorem and some applications. Math. Z., 72 (1959), 192204.CrossRefGoogle Scholar
11. Richert, H. E.. Zur Abschätzung der Riemannschen Zetafunktion in der Nähe der Vertikalen σ = 1. Math. Ann., 169, 97101.CrossRefGoogle Scholar
12. Sokolovsky, A. V.. A theorem on the zeros of Dedekinds zeta-functions and the distance between ‘neighbouring’ prime ideals. Acta Arith., 13 (1968), 321334.Google Scholar
13. Titchmarsh, E. C.. The theory of the Riemann Zeta-Function (Oxford, 1951).Google Scholar
14. Urbialis, J.. Distribution of algebraic primes. Liet. Mat. sb., 5 (1965), 504516.Google Scholar
15. Walfisz, A.. Weylische Exponentialsummen in der Neueren Zahlentheorie (VEB Deutscher Verlag, Berlin, 1963).Google Scholar