

A278790


Number of real cubic fields with discriminant <= 10^n.


2



0, 2, 27, 382, 4804, 54600, 592922, 6248290, 64659361, 661448081, 6715824025
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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), 12131237.
Manjul Bhargava, Arul Shankar, Jacob Tsimerman, On the DavenportHeilbronn theorems and second order terms, Invent. math. 193:2 (2013) 439499.
David P. Roberts, Density of cubic field discriminants, Math. Comp. 70 (2001), 16991705.


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



