Numerical secondary terms in a Cohen-Lenstra conjecture on real quadratic fields. (English) Zbl 1458.11158

Summary: In [Lect. Notes Math. 1068, 33–62 (1984; Zbl 0558.12002)], H. Cohen and H. W. Lenstra jun. made a number of conjectures regarding the class groups of quadratic fields. In particular, they predicted the proportion of real quadratic fields with class number divisible by an odd prime. We numerically investigate the difference between reality and these predictions. Using 4 million data points, we perform a curve fitting of the difference with a monomial term and demonstrate that there is reason to believe the term can be effectively approximated within the scope of our data set for odd primes less than 30. We use cross-validation to show that including our monomial term as a secondary term to the original conjecture reduces the overall error.


11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11Y35 Analytic computations


Zbl 0558.12002
Full Text: DOI


[1] 10.1007/s00222-012-0433-0 · Zbl 1294.11191
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[9] 10.1215/00127094-2371752 · Zbl 1294.11192
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