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2003 New Computations Concerning the Cohen-Lenstra Heuristics
Hugh Williams, Herman te Riele
Experiment. Math. 12(1): 99-113 (2003).


Let h(p) denote the class number of the real quadratic field formed by adjoining $\sqp$} where p is a prime, to the rationals. The Cohen-Lenstra heuristics suggest that the probability that $h(p)=k$ (a given odd positive integer) is given by $Cw(k)/k$, where C is an explicit constant and w(k) is an explicit arithmetic function. For example, we expect that about 75.45 % of the values of h(p) are 1, 12.57% are 3, and 3.77% are 5. Furthermore, a conjecture of Hooley states that

$H(x):=\sum_{p\le x}h(p)\sim x/8,$

where the sum is taken over all primes congruent to 1 modulo 4. In this paper, we develop some fast techniques for evaluating h(p) where p is not very large and provide some computational results in support of the Cohen-Lenstra heuristics. We do this by computing h(p) for all p ($\equiv1\bmod{4}$) and $p<2\cdot10^{11}$. We also tabulate H(x) up to $2\cdot10^{11}$.


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Hugh Williams. Herman te Riele. "New Computations Concerning the Cohen-Lenstra Heuristics." Experiment. Math. 12 (1) 99 - 113, 2003.


Published: 2003
First available in Project Euclid: 29 September 2003

zbMATH: 1050.11096
MathSciNet: MR2002677

Primary: 11R29
Secondary: 11Y40

Keywords: Cohen-Lenstra heuristics , Distribution of class numbers , Hooley's conjecture

Rights: Copyright © 2003 A K Peters, Ltd.


Vol.12 • No. 1 • 2003
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