The Pólya-Vinogradov inequality for totally real algebraic number fields

Perfect Forms over Totally Real Number Fields

A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronöı and later generalized by Koecher to arbitrary number fields. One knows that up to a natural “change of variables” equivalence, there are only finitely many perfect f...

Units in families of totally complex algebraic number fields

Multidimensional continued fraction algorithms associated with GL n (Z K), where Z k is the ring of integers of an imaginary quadratic field K, are introduced and applied to find systems of fundamental units in families of totally complex algebraic number fields of degrees four, six, and eight. 1. Introduction. Let F be an algebraic number field of degree n. There exist exactly n field embeddin...

On Shintani’s ray class invariant for totally real number fields

We introduce a ray class invariant X(C) for a totally real field, following Shintani’s work in the real quadratic case. We prove a factorization formula X(C) = X1(C) · · ·Xn(C) where each Xi(C) corresponds to a real place (Theorem 3.5). Although this factorization depends a priori on some choices (especially on a cone decomposition), we can show that it is actually independent of these choices ...

Gross–Stark units for totally real number fields

19 INTEGERS 11 A ( 2011 ) Proceedings of Integers Conference 2009 REMARKS ON THE PÓLYA – VINOGRADOV INEQUALITY

We establish a numerically explicit version of the Pólya–Vinogradov inequality for the sum of values of a Dirichlet character on an interval. While the technique of proof is essentially that of Landau from 1918, the result we obtain has better constants than in other numerically explicit versions that have been found more recently. – Dedicated to Mel Nathanson on his 65th birthday