Motivation: Our textbook discusses and even proves many properties of R, the field of real numbers; but it doesn’t define it. I felt that it would be rather awkward to discuss real numbers without knowing what they were and I decided to write some notes on the construction of R. The approach I am following is called ‘Dedekind cut’, discovered by a German mathematician, Richard Dedekind (1831-19...

— We give an explicit upper bound for the residue at s = 1 of the Dedekind zeta function of a totally real number field K for which ζK(s)/ζ(s) is entire. Notice that this is conjecturally always the case, and that it holds true if K/Q is normal or if K is cubic. Résumé (Bornes supérieures explicites pour les résidus en s = 1 des fonctions zêta de Dedekind de corps de nombres totalement réels) N...

We give explicit, polynomial–time computable formulas for the number of integer points in any two– dimensional rational polygon. A rational polygon is one whose vertices have rational coordinates. We find that the basic building blocks of our formulas are Dedekind–Rademacher sums, which are polynomial–time computable finite Fourier series. As a by–product we rederive a reciprocity law for these...

‎In this paper‎, ‎we introduce a new sum‎ ‎analogous to Gauss sum‎, ‎then we use the properties of the‎ ‎classical Gauss sums and analytic method to study the hybrid mean‎ ‎value problem involving this new sums and Dedekind sums‎, ‎and‎ ‎give an interesting identity for it.

Let z be a real quadratic irrational. We compare the asymptotic behavior of Dedekind sums S(pk, qk) belonging to convergents pk/qk of the regular continued fraction expansion of z with that of Dedekind sums S(sj/tj) belonging to convergents sj/tj of the negative regular continued fraction expansion of z. Whereas the three main cases of this behavior are closely related, a more detailed study of...

The ring Z consists of the integers of the field Q, and Dedekind takes the theory of unique factorization in Z to be clear and well understood. The problem is that unique factorization can fail when one considers the integers in a finite extension of the rationals, Q(α). Kummer showed that when Q(α) is a cyclotomic extension (i.e. α is a primitive pth root of unity for a prime number p), one ca...