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A GENERALISED KUMMER'S CONJECTURE

Published online by Cambridge University Press:  25 August 2010

M. J. R. MYERS
M. J. R. MYERS*
Affiliation:
Department of Mathematics and Statistics, Calvin College, Grand Rapids, MI 49546, USA e-mail: [email protected]
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Abstract

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Kummer's conjecture predicts the rate of growth of the relative class numbers of cyclotomic fields of prime conductor. We extend Kummer's conjecture to cyclotomic fields of conductor n, where n is any natural number. We show that the Elliott–Halberstam conjecture implies that this generalised Kummer's conjecture is true for almost all n but is false for infinitely many n.

Keywords

Primary 11R18Secondary 11M20
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 3 , September 2010 , pp. 453 - 472
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Ankeny, N. C. and Chowla, S., The class number of the cyclotomic field. Canad. J. Math. 3 (1951), 486–494.CrossRefGoogle Scholar
2.Apostol, T. M., Introduction to analytic number theory (Springer-Verlag, New York, 1995).Google Scholar
3.Granville, A., On the size of the first factor of the class number of a cyclotomic field, Invent. Math. 100 (1990), 321–338.CrossRefGoogle Scholar
4.Halberstam, H. and Richert, H. E., Sieve methods (Academic Press, London, 1974).Google Scholar
5.Hardy, G. H. and Ramanujan, S., The normal number of prime factors of a number n, Q. J. Math. 48 (1917), 76–92.Google Scholar
6.Hooley, C., On the Brun–Titchmarsh theorem, II, Proc. Lond. Math. Soc. (3) 30 (1975), 114–128.CrossRefGoogle Scholar
7.Kummer, E. E., Collected papers (Springer-Verlag, New York, 1975).Google Scholar
8.Lu, Y. and Zhang, W., On the Kummer conjecture, Acta Arith. 131 (2008), 87–102.CrossRefGoogle Scholar
9.Murty, M. R. and Petridis, Y. N., On Kummer's conjecture, J. Number Theory 90 (2001), 294–303.CrossRefGoogle Scholar
10.Myers, M. J. R., A prime power variation of Kummer's conjecture, J. Ramanujan Math. Soc. (submitted).Google Scholar
11.Washington, L. C., Introduction to cyclotomic fields (Springer-Verlag, New York, 1982).CrossRefGoogle Scholar