New computations concerning the Cohen-Lenstra heuristics

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 $2 imes10^{11}$.