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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
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References:

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