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A278791 Number of complex cubic fields with discriminant >= -10^n. 2
0, 7, 127, 1520, 17041, 182417, 1905514, 19609185, 199884780, 2024660098, 20422230540 (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 complex cubic fields with discriminant >= -X is asymptotic to X/(4*zeta(3)) = (0.207976...)*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. A023679, A278790.

Sequence in context: A053716 A088550 A255954 * A064754 A267249 A025166

Adjacent sequences:  A278788 A278789 A278790 * A278792 A278793 A278794

KEYWORD

nonn,more

AUTHOR

Christopher E. Thompson, Nov 28 2016

STATUS

approved

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Last modified October 20 17:50 EDT 2019. Contains 328268 sequences. (Running on oeis4.)