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Density of cubic field discriminants


Author: David P. Roberts
Journal: Math. Comp. 70 (2001), 1699-1705
MSC (2000): Primary 11N56, 11R16
DOI: https://doi.org/10.1090/S0025-5718-00-01291-6
Published electronically: October 18, 2000
Erratum: Math. Comp. 41 (1983), 775-778.
Erratum: Math. Comp. 36 (1981), 316-317.
Erratum: Math. Comp. 36 (1981), 315-316.
MathSciNet review: 1836927
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Abstract:

In this paper we give a conjectural refinement of the Davenport-Heilbronn theorem on the density of cubic field discriminants. Our refinement is plausible theoretically and agrees very well with computational data.


References [Enhancements On Off] (What's this?)

  • 1. I. O. Angell, A table of totally real cubic fields, Math. Comput. 30 (1976), 184-187. MR 53:5528
  • 2. K. Belabas, Crible et $3$-rang des corps quadratiques, Ann. Inst. Fourier (Grenoble), 46 (1996), 909-949. MR 98b:11112
  • 3. -, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237. MR 97m:11159
  • 4. H. Cohn, The density of abelian cubic fields, Proc. Amer. Math. Soc. 5 (1954), 476-477. MR 16:222a
  • 5. B. Datskovsky and D. J. Wright, The adelic zeta function associated with the space of binary cubic forms, II: local theory, J. Reine Angew. Math. 367 (1986), 27-75. MR 87m:11034
  • 6. -, Density of discriminants of cubic extensions, J. Reine Angew. Math. 386 (1988) 116-138. MR 90b:11112
  • 7. H. Davenport, On the class number of binary cubic forms, I & II., J. London Math. Soc. 26 (1951), 183-198 (Corrigendum, ibid 27 (1952) 512). MR 13:323e, MR 13:323f
  • 8. H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields (II), Proc. Roy. Soc. Lond. A 322 (1971) 405-420 MR 58:10816
  • 9. G. W. Fung and H. G. Williams, On the computation of complex cubic fields, with discriminant $D \geq {-10^6}$, Math. Comp. 55 (1990), 313-325. Errata Math. Comp. 63 (1994), 433. MR 94i:11106
  • 10. P. Llorente and J. Quer, On totally real cubic fields with discriminant $d < 10^7$, Math. Comp. 50 (1988), 581-594. MR 89g:11099
  • 11. Y. Ohno, A conjecture on coincidence among the zeta functions associated with the space of binary cubic forms, Amer. J. of Math. 119 (1997), 1083-1094. MR 98k:11037
  • 12. M. Sato and T. Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. 100 (1974), 131-170. MR 49:8969
  • 13. D. Shanks, Review of [1], Math. Comp. 30, (1976), 670-673.
  • 14. -, A survey of quadratic, cubic and quartic algebraic number fields (from a computational point of view), Cong. Numer. 17 (1976), 15-40. MR 56:11951
  • 15. T. Shintani, On Dirichlet series whose coefficients are class numbers of integral binary cubic forms, J. Math. Soc. Japan, 24 (1972), 132-188. MR 44:6619

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Additional Information

David P. Roberts
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
Address at time of publication: Division of Science and Mathematics, University of Minnesota-Morris, Morris, Minnesota 56267
Email: roberts@mrs.umn.edu

DOI: https://doi.org/10.1090/S0025-5718-00-01291-6
Received by editor(s): April 20, 1999
Received by editor(s) in revised form: January 6, 2000
Published electronically: October 18, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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