Dedekind sums: a combinatorial-geometric viewpoint

The literature on Dedekind sums is vast. In this expository paper we show that there is a common thread to many generalizations of Dedekind sums, namely through the study of lattice point enumeration of rational polytopes. In particular, there are some natural finite Fourier series which we call Fourier-Dedekind sums, and which form the building blocks of the number of partitions of an integer from a finite set of positive integers. This problem also goes by the name of the `coin exchange problem'. Dedekind sums have enjoyed a resurgence of interest recently, from such diverse fields as topology, number theory, and combinatorial geometry. The Fourier-Dedekind sums we study here include as special cases generalized Dedekind sums studied by Berndt, Carlitz, Grosswald, Knuth, Rademacher, and Zagier. Our interest in these sums stems from the appearance of Dedekind's and Zagier's sums in lattice point count formulas for polytopes. Using some simple generating functions, we show that generalized Dedekind sums are natural ingredients for such formulas. As immediate `geometric' corollaries to our formulas, we obtain and generalize reciprocity laws of Dedekind, Zagier, and Gessel. Finally, we prove a polynomial-time complexity result for Zagier's higher-dimensional Dedekind sums.Comment: 11 page

Explicit and efficient formulas for the lattice point count in rational polygons using Dedekind-Rademacher sums

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 sums due to Gessel, which generalizes the reciprocity law for the classical Dedekind sums. In addition, our approach shows that Gessel's reciprocity law is a special case of the one for Dedekind-Rademacher sums, due to Rademacher.Comment: 16 pages, updated journal referenc

On sums of powers of cosecs

The finite sums of powers of cosecs occur in numerous situations, both physical and mathematical, examples being the Casimir effect, Renyi entropy, Verlinde's formula and Dedekind sums. I here present some further discussion which consists mainly of a reprise of early work by H.M.Jeffery in 1862-64 which has fallen by the wayside and whose results are being reproduced up to the present day. The motivation is partly historical justice and partly that, because of the continuing appearance of the sums, his particular methods deserve re--exposure. For example, simple trigonometric generating functions are found and these have a field theoretic, Green function significance and I make a few comments in the topic of R\'enyi entropies.Comment: 14 page

Evaluation of Dedekind sums, Eisenstein cocycles, and special values of L-functions

We define certain higher-dimensional Dedekind sums that generalize the classical Dedekind-Rademacher sums, and show how to compute them effectively using a generalization of the continued-fraction algorithm. We present two applications. First, we show how to express special values of partial zeta functions associated to totally real number fields in terms of these sums via the Eisenstein cocycle introduced by the second author. Hence we obtain a polynomial-time algorithm for computing these special values. Second, we show how to use our techniques to compute certain special values of the Witten zeta-function, and compute some explicit examples.Comment: fixed typo

Computing special values of partial zeta functions

We discuss computation of the special values of partial zeta functions associated to totally real number fields. The main tool is the \emph{Eisenstein cocycle} $\Psi$, a group cocycle for $GL_{n} (\Z )$; the special values are computed as periods of $\Psi$, and are expressed in terms of generalized Dedekind sums. We conclude with some numerical examples for cubic and quartic fields of small discriminant.Comment: 10 p

A note on the q-Dedekind-type Daehee-Changhee sums with weight alpha arising from modified q-Genocchi polynomials with weight alpha

In the present paper, our objective is to treat a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order Dedekind-type sums with weight in connection with modified q-Genocchi polynomials with weight alpha by using p-adic invariant q-integral on Zp.Comment: 6 pages, submitte

Generalized Dedekind sums and equidistribution mod 1

Dedekind sums are well-studied arithmetic sums, with values uniformly distributed on the unit interval. Based on their relation to certain modular forms, Dedekind sums may be defined as functions on the cusp set of $SL(2,\mathbb{Z})$. We present a compatible notion of Dedekind sums, which we name Dedekind symbols, for any non-cocompact lattice $\Gamma<SL(2,\mathbb{R})$, and prove the corresponding equidistribution mod 1 result. The latter part builds up on a paper of Vardi, who first connected exponential sums of Dedekind sums to Kloosterman sums.Comment: 22 page

Transformation laws for generalized Dedekind sums associated to Fuchsian groups

We establish transformation laws for generalized Dedekind sums associated to the Kronecker limit function of non-holomorphic Eisenstein series and their higher-order variants. These results apply to general Fuchsian groups of the first kind, and examples are provided in the cases of the Hecke triangle groups, the Hecke congruence groups $\Gamma_0(N)$, and the non-congruence arithmetic groups $\Gamma_0(N)^+$.Comment: 18 page

Covering RR-modules by proper submodules

A classical problem in the literature seeks the minimal number of proper subgroups whose union is a given finite group. A different question, with applications to error-correcting codes and graph colorings, involves covering vector spaces over finite fields by (minimally many) proper subspaces. We unify these questions by working over an arbitrary commutative ring $R$, and studying the smallest cardinal number $\aleph$, possibly infinite, such that a given $R$-module is a union of $\aleph$-many proper submodules. In this note: (1) We characterize when $\aleph$ is a finite cardinal; this parallels for modules a 1954 result of Neumann. (2) We also compute the covering (cardinal) number of a finitely generated module over a quasi-local ring. (3) As a variant, we compute the covering number of an arbitrary direct sum of cyclic monoids. Our proofs are self-contained.Comment: 9 pages, no figures. Major rewriting and update

Macdonald's solid-angle sum for real dilations of rational polygons

The solid-angle sum $A_{\mathcal{P}} (t)$ of a rational polytope ${\mathcal{P}} \subset \mathbb{R}^d$, with $t \in \mathbb{Z}$ was first investigated by I.G. Macdonald. Using our Fourier-analytic methods, we are able to establish an explicit formula for $A_{\mathcal{P}} (t)$, for any real dilation $t$ and any rational polygon ${\mathcal{P}} \subset \mathbb{R}^2$. Our formulation sheds additional light on previous results, for lattice-point enumerating functions of triangles, which are usually confined to the case of integer dilations. Our approach differs from that of Hardy and Littlewood in 1992, but offers an alternate point of view for enumerating weighted lattice points in real dilations of real triangles.Comment: 20 pages, 4 figure