New Computations Concerning the Cohen-Lenstra Heuristics

Abstract

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 C _w (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

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. 1011. We also tabulate H(x) up to 2.1011.

Keywords:

• Distribution of class numbers
• Cohen-Lenstra heuristics
• Hooley's conjecture