Let $h(p)$ denote the class number of the real quadratic field formed by adjoining $sqrt{p}$, 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 $75.45$ ofthe values of $ h(p)$ are 1, $12.57$ are 3, and $3.77$ are 5.Furthermore, a conjecture of Hooley states that$$H(x):=sum_{ple x}h(p)sim x/8 mbox{~as~} x ightarrowinfty,$$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 ofthe Cohen-Lenstra heuristics. We do this by computing $h(p)$ for all $p$($equiv1 mod{4}$) and $p<2 imes10^{11}$. We also tabulate $H(x)$up to $2 imes10^{11}$.

Numerical Algorithms and Problems (acm F.2.1)
Class numbers, class groups, discriminants (msc 11R29), Algebraic number theory computations (msc 11Y40)
Life Sciences (theme 5), Energy (theme 4)
CWI
Modelling, Analysis and Simulation [MAS]
Scientific Computing

Williams, H.C, & te Riele, H.J.J. (2002). New computations concerning the Cohen-Lenstra heuristics. Modelling, Analysis and Simulation [MAS]. CWI.