A278790 Number of real cubic fields with discriminant <= 10^n.

0, 2, 27, 382, 4804, 54600, 592922, 6248290, 64659361, 661448081, 6715824025

Offset: 1

Author

Christopher E. Thompson, Nov 28 2016

Keywords

• nonn
• more

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

• 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.