Let h(p) denote the class number of the real quadratic field formed by adjoining √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% 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):= Σh(p) ∼ x/8 as x → oo, p≤x 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 (≡ 1 mod 4) and p < 2 × 10 11 . We also tabulate H(x) up to 2 × 10 11 . 2000 Mathematics Subject Classification: Primary 11R29. Secondary 11Y40. 1998 ACM Computing Classification System: F.2.1.
['Hugh C. Williams', 'Herman J. J. te Riele']
New computations concerning the Cohen-Lenstra heuristics