A278791 Number of complex cubic fields with discriminant >= -10^n.

0, 7, 127, 1520, 17041, 182417, 1905514, 19609185, 199884780, 2024660098, 20422230540

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 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

• 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.