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Number of complex cubic fields with discriminant >= -10^n.
2

%I #16 Dec 08 2016 18:13:54

%S 0,7,127,1520,17041,182417,1905514,19609185,199884780,2024660098,

%T 20422230540

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

%C 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).

%C 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.

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

%H Karim Belabas, <a href="https://doi.org/10.1090/S0025-5718-97-00846-6">A fast algorithm to compute cubic fields</a>, Math. Comp. 66 (1997), 1213-1237.

%H Manjul Bhargava, Arul Shankar, Jacob Tsimerman, <a href="https://doi.org/10.1007/s00222-012-0433-0">On the Davenport-Heilbronn theorems and second order terms</a>, Invent. math. 193:2 (2013) 439-499.

%H David P. Roberts, <a href="https://doi.org/10.1090/S0025-5718-00-01291-6">Density of cubic field discriminants</a>, Math. Comp. 70 (2001), 1699-1705.

%Y Cf. A023679, A278790.

%K nonn,more

%O 1,2

%A _Christopher E. Thompson_, Nov 28 2016