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Assuming the Generalized Riemann Hypothesis, Bach has shown that the ideal class group C K of a number field K can be generated by the prime ideals of K having norm smaller than 12 log |Discriminant(K)| 2. This result is essential for the computation of the class group and units of K by Buchmann's algorithm, currently the fastest known. However, once C K(More)
We show that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold. The next smallest one is the Meyerhoff manifold. A hyperbolic 3-manifold is a 3-manifold admitting a complete Riemannian metric all of whose sectional curvatures are −1. The universal cover of such a manifold can therefore be identified with the hyperbolic 3-space,(More)
Assuming the Generalized Riemann Hypothesis , Bach has shown that one can calculate the residue of the Dedekind zeta function of a number field K by a clever use of the splitting of primes p < X, with an error asymptotically bounded by 8.33 log ∆ K /(√ X log X), where ∆ K is the absolute value of the discriminant of K. Guided by Weil's explicit formula and(More)
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