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Sieve and 3-rank of quadratic fields. (Crible et 3-rang des corps quadratiques.) (French) Zbl 0853.11088
Summary: Call \(h^*_3(\Delta)\) the number of cube roots of unity in the class group of \(\mathbb{Q}(\sqrt\Delta)\), where \(\Delta\) is a fundamental discriminant. Davenport and Heilbronn computed the mean value of these numbers when \(\Delta\) tends to \(\pm\infty\). The author gives a general geometric argument yielding an explicit bound for the error term, with the additional possibility of restricting \(\Delta\) to arithmetic progressions. Sieve techniques then produce results about the 3-parts of the groups \(\text{Cl}(\mathbb{Q}(\sqrt{\Delta}))\), where \(P_k\) is an almost-prime of order \(k\). In this way, one controls simultaneously both the 2-rank and the 3-rank of the class group \(\text{Cl}(\mathbb{Q}(\sqrt{\Delta}))\). As a special case, the author gives a bound for the mean 3-rank of the \(\mathbb{Q} (\sqrt{\pm p})\), where \(p\) is prime.

11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11N36 Applications of sieve methods
11P21 Lattice points in specified regions
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