Dedekind sums in function fields

Okada (J Number Theory, 130:1750–1762, 2010) introduced Dedekind sums associated to a certain A-lattice, and established the reciprocity law. In this paper, we introduce Dedekind sums for arbitrary A-lattice and establish the reciprocity law for them. We next introduce higher dimensional Dedekind sums for any A-lattice. These Dedekind sums are analogues of Zagier’s higher dimensional Dedekind sums. We discuss the reciprocity law, rationality and characterization of these sums.     

Dedekind sums with a parameter in finite fields

We first introduce the multiple Dedekind–Rademacher sum with a parameter in finite fields and establish its reciprocity law. We then construct an analog of the higher-dimensional Apostol–Dedekind sums, and establish their reciprocity laws using the parameterized Dedekind sum.     

The Dedekind symbol associated with the Eisenstein series of weight two

Dedekind symbols generalize the classical Dedekind sums (symbols). These symbols are determined uniquely, up to additive constants, by their reciprocity laws. For k ≧ 2, there is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws of degree 2k − 2 and the space of modular forms of weight 2k for the full modular group However, this is not the case when k = 1 as there is no modular form of weight two; nevertheless, there exists a unique (up to a scalar multiple) quasi-modular form (Eisenstein series) of weight two. The purpose of this note is to define the Dedekind symbol associated with this quasi-modular form, and to prove its reciprocity law. Furthermore we show that the odd part of this Dedekind symbol is nothing but a scalar multiple of the classical Dedekind sum. This gives yet another proof of the reciprocity law for the classical Dedekind sum in terms of the quasi-modular form.Received: 13 September 2004     

Dedekind symbols with plus reciprocity laws

Dedekind symbols are generalizations of the classical Dedekind sums (symbols), and the symbols are determined uniquely by their reciprocity laws, up to an additive constant. For Dedekind symbols D and F, we can consider two kinds of reciprocity laws: D(p,q)−D(q,−p)=R(p,q) and F(p,q)+F(q,−p)=T(p,q). The first type, which we call minus reciprocity laws, have been studied extensively. On the contrary, the second type, which we call plus reciprocity laws, have not yet been investigated. In this note we study fundamental properties of Dedekind symbols with plus reciprocity law F(p,q)+F(q,−p)=T(p,q). We will see that there is a fundamental difference between Dedekind symbols with minus and plus reciprocity laws.     

Degenerate and character Dedekind sums

In this paper, we study on two subjects. We first construct degenerate analogues of Dedekind sums in the sense of Apostol, Carlitz and Takács, and prove the corresponding reciprocity formulas. Secondly, we define generalized Dedekind character sums, which are explicit extensions of Berndt's definition, and prove the reciprocity laws. From the derived reciprocity laws, we obtain Berndt's reciprocity laws as special cases.     

Explicit and Efficient Formulas for the Lattice Point Count in Rational Polygons Using Dedekind—Rademacher Sums

Abstract. 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.     

Dedekind–Carlitz polynomials as lattice-point enumerators in rational polyhedra

We study higher-dimensional analogs of the Dedekind–Carlitz polynomials , where u and v are indeterminates and a and b are positive integers. Carlitz proved that these polynomials satisfy the reciprocity law from which one easily deduces many classical reciprocity theorems for the Dedekind sum and its generalizations. We illustrate that Dedekind–Carlitz polynomials appear naturally in generating functions of rational cones and use this fact to give geometric proofs of the Carlitz reciprocity law and various extensions of it. Our approach gives rise to new reciprocity theorems and computational complexity results for Dedekind–Carlitz polynomials, a characterization of Dedekind–Carlitz polynomials in terms of generating functions of lattice points in triangles, and a multivariate generalization of the Mordell–Pommersheim theorem on the appearance of Dedekind sums in Ehrhart polynomials of 3-dimensional lattice polytopes. Research of Haase supported by DFG Emmy Noether fellowship HA 4383/1. We thank Robin Chapman, Eric Mortenson, and an anonymous referee for helpful comments.     

Generalized Dedekind symbols associated with the Eisenstein series

We have shown recently that the space of modular forms, the space of generalized Dedekind sums, and the space of period polynomials are all isomorphic. In this paper, we will prove, under these isomorphisms, that the Eisenstein series correspond to the Apostol generalized Dedekind sums, and that the period polynomials are expressed in terms of Bernoulli numbers. This gives us a new more natural proof of the reciprocity law for the Apostol generalized Dedekind sums. Our proof yields as a by-product new polylogarithm identities.

     

Higher dimensional Dedekind sums in finite fields

We introduce Dedekind sums of a new type defined over finite fields. These are similar to the higher dimensional Dedekind sums of Zagier. The main result is the reciprocity law for them.     

Reciprocity formulae for multiple Dedekind–Rademacher sums

We introduce multiple Dedekind–Rademacher sums, in terms of values of Bernoulli functions, that generalize the classical Dedekind–Rademacher sums. The aim of this paper is to give and prove a reciprocity law for these sums. The main theorem presented in this paper contains all previous results in the literature about Dedekind–Rademacher sums.     

A new generalization of Hardy–Berndt sums

In this paper, we construct a new generalization of Hardy–Berndt sums which are explicit extensions of Hardy–Berndt sums. We express these sums in terms of Dedekind sums s _r (h, k : x, y|λ) with x?=?y?=?0 and obtain corresponding reciprocity formulas.     

The Pick Theorem and the Proof of the Reciprocity Law for Dedekind Sums

This paper is to provide some new generalizations of the Pick Theorem. We first derive a point-set version of the Pick Theorem for an arbitrary bounded lattice polyhedron. Then, we use the idea of a weight function of [2] to obtain a weighted version. Other Pick type theorems known to the author for the integral lattice Z² are reduced to some special cases of this generalization. Finally, using an idea of Ehrhart [6] and the Pick Theorem, we give a direct proof of the reciprocity law for Dedekind sums. The ideas and methods presented here may be pushed to higher dimensions.AMS Subject Classification: 52C05, 11H06, 57N05, 57N15, 57N35.     

Dedekind symbols with polynomial reciprocity laws

A Dedekind symbol is a generalization of the classical Dedekind symbol (sum). A Dedekind symbol is characterized by its reciprocity law. Dedekind symbols with polynomial reciprocity laws are of special interest and importance, as such symbols are known to correspond bijectively to cusp forms for the full modular group, and to period polynomials. However, explicit forms of such Dedekind symbols are not yet known. In this article we construct Dedekind symbols explicitly by means of Poincaré series, and then show that these symbols satisfy polynomial reciprocity laws and that they form a spanning set for the space of Dedekind symbols with polynomial reciprocity laws. That is, we show that any Dedekind symbol with polynomial reciprocity law can be expressed as a linear combination of these symbols.Mathematics Subject Classification (2000): 11F20; 11F11, 33E05The author wishes to thank Professor N. Yui for her helpful advice.     

The elliptic Apostol–Dedekind sums generate odd Dedekind symbols with Laurent polynomial reciprocity laws

Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a new elliptic analogue of the Apostol–Dedekind sums. Then we will show that the newly defined sums generate all odd Dedekind symbols with Laurent polynomial reciprocity laws. Our construction is based on Machide’s result (J Number Theory 128:1060–1073, 2008) on his elliptic Dedekind–Rademacher sums. As an application of our results, we discover Eisenstein series identities which generalize certain formulas by Ramanujan (Collected Papers of Srinivasa Ramanujan, pp. 136–162. AMS Chelsea Publishing, Providence, 2000), van der Pol (Indag Math 13:261–271, 272–284, 1951), Rankin (Proc R Soc Edinburgh Sect A 76:107–117, 1976) and Skoruppa (J Number Theory 43:68–73, 1993).     

On generalized Dedekind sums attached to Dirichlet characters

Generalized reciprocity formulas and Dedekind-Petersson-Knopp-type formulas are given to generalized Dedekind sums attached to Dirichlet characters, defined on a certain congruence subgroup of SL₂($\text{Z}$). In addition, these formulas are respectively construed as transformational and eigen properties of those sums redefined on a certain set of cusps.     

Applications aux sommes elliptiques d'Apostol–Dedekind–Zagier

In this Note, we give two applications to our work [Bayad, C. R. Acad. Sci. Paris, Ser. I 339 (2004); DOI: 10.1016/j.crma.2004.07.018] concerning multiple elliptic Apostol–Dedekind–Zagier sums. These elliptic sums are defined by means of certain Jacobi modular forms of two variables $D_{\tau }(z\text{;}\phi )$. When $\text{Im}(\tau )\to \infty$, these elliptic sums give the classical Apostol–Dedekind–Zagier multiple sums [Apostol, Duke Math. J. 17 (1950) 147–157, Pacific. J. Math 2 (1952) 1–9; Zagier, Math. Ann, 202 (1973) 149–172]. We give a reciprocity law for these sums. To cite this article: A. Bayad, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

q-Dedekind type sums related to q-zeta function and basic L-series

The main purpose of this paper is to define new generating functions. By applying the Mellin transformation formula to these generating functions, we define q-analogue of Riemann zeta function, q-analogue Hurwitz zeta function, q-analogue Dirichlet L-function and two-variable q-L-function. In particular, by using these generating functions, we will construct new generating functions which produce q-Dedekind type sums and q-Dedekind type sums attached to Dirichlet character. We also give the relations between these sums and Dedekind sums. Furthermore, by using *-product which is given in this paper, we will give the relation between Dedekind sums and q-L function as well.     

Values of zeta functions at negative integers, Dedekind sums and toric geometry

We study relations among special values of zeta functions, invariants of toric varieties, and generalized Dedekind sums. In particular, we use invariants arising in the Todd class of a toric variety to give a new explicit formula for the values of the zeta function of a real quadratic field at nonpositive integers. We also express these invariants in terms of the generalized Dedekind sums studied previously by several authors. The paper includes conceptual proofs of these relations and explicit computations of the various zeta values and Dedekind sums involved.