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A278790 Number of real cubic fields with discriminant <= 10^n. 2
0, 2, 27, 382, 4804, 54600, 592922, 6248290, 64659361, 661448081, 6715824025 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Belabas invented an algorithm to identify all cubic fields with a discriminant bounded by X in essentially linear time, and computed the above values up to a(11).

The number of real cubic fields with discriminant <= X is asymptotic to X/(12*zeta(3)) = (0.069325...)*X. The second order term was conjectured by Roberts to be a known constant times X^{5/6}, and this was subsequently proved by Bhargava et al.

REFERENCES

Henri Cohen, Advanced Topics in Computational Number Theory, Springer, 2000, p. 426 (and Chapter 8 more generally).

LINKS

Table of n, a(n) for n=1..11.

Karim Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 1213-1237.

Manjul Bhargava, Arul Shankar, Jacob Tsimerman, On the Davenport-Heilbronn theorems and second order terms, Invent. math. 193:2 (2013) 439-499.

David P. Roberts, Density of cubic field discriminants, Math. Comp. 70 (2001), 1699-1705.

CROSSREFS

Cf. A006832, A278791.

Sequence in context: A091709 A083384 A121971 * A277832 A221674 A240701

Adjacent sequences:  A278787 A278788 A278789 * A278791 A278792 A278793

KEYWORD

nonn,more

AUTHOR

Christopher E. Thompson, Nov 28 2016

STATUS

approved

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Last modified October 21 22:47 EDT 2021. Contains 348160 sequences. (Running on oeis4.)