## New computations concerning the Cohen-Lenstra heuristics.(English)Zbl 1050.11096

Consider the quadratic number field $$\mathbb Q(\sqrt{p}\,)$$ with class number $$h(p)$$, where $$p \equiv 1 \bmod 4$$ is a prime. The Cohen-Lenstra heuristics [H. Cohen and H. W. Lenstra, Lect. Notes Math. 1052, 26–36 (1984; Zbl 0532.12008)] predicts e.g. that about 75.45 of such fields have class number $$1$$. The authors provide new numerical evidence for these conjectures by computing the class number $$h(p)$$ for all such $$p < 2 \cdot 10^{11}$$; this is accomplished by developing a fast technique for computing $$h(p)$$.
In addition, the authors test the conjecture of C. Hooley [J. Reine Angew. Math. 353, 98–131 (1984; Zbl 0539.10019)] that $$H(x) = \sum{p \leq x} h(p) \sim x/8$$. They find that if $$8 H(x)/x$$ converges, then it does so very slowly.

### MSC:

 11R29 Class numbers, class groups, discriminants 11Y40 Algebraic number theory computations 11R11 Quadratic extensions

### Citations:

Zbl 0532.12008; Zbl 0539.10019
Full Text:

### References:

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