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...

We describe a new series of identities, which hold for certain general theta series, in two completely independent variables. We provide explicit examples of these identities involving the Dedekind eta function, Jacobi theta functions, and various theta functions of Ramanujan. Introduction Let z ∈ H = {x+ yi : x, y ∈ R, y > 0} and for each x ∈ R set q = exp(2πixz) and e(x) = exp(2πix). The Dede...