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.     

A generating function of higher-dimensional Apostol-Zagier sums and its reciprocity law

We introduce higher-dimensional Dedekind sums with a complex parameter z, generalizing Zagier's higher-dimensional Dedekind sums. The sums tend to Zagier's higher-dimensional Dedekind sums as z→∞. We show that the sums turn out to be generating functions of higher-dimensional Apostol-Zagier sums which are defined to be hybrids of Apostol's sums and Zagier's sums. We prove reciprocity law for the sums. The new reciprocity law includes reciprocity formulas for both Apostol and Zagier's sums as its special case. Furthermore, as its application we obtain relations between special values of Hurwitz zeta function and Bernoulli numbers, as well as new trigonometric identities.     

An identity involving Dedekind sums and generalized Kloosterman sums

The various properties of classical Dedekind sums S(h, q) have been investi-gated by many authors. For example, Yanni Liu and Wenpeng Zhang: A hybrid mean value related to the Dedekind sums and Kloosterman sums, Acta Mathematica Sinica, 27 (2011), 435–440 studied the hybrid mean value properties involving Dedekind sums and generalized Kloosterman sums K(m, n, r; q). The main purpose of this paper, is using the analytic methods and the properties of character sums, to study the computational problem of one kind of hybrid mean value involving Dedekind sums and generalized Kloosterman sums, and give an interesting identity.     

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.     

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.

     

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.

     

广义Dedekind和与Hardy和

本文给出了广义Dedekind和与Hardy和的定义,研究了广义Dedekind和的算术性质,并把Hardy和表示成广义Dedekind和的形式．提出了广义Subrahmanyam等式和Knopp定理,并给出了证明．     

On the mean value of dedekind sum weighted by the quadratic gauss sum

Various properties of classical Dedekind sums S(h, q) have been investigated by many authors. For example, Wenpeng Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordx, 8 (1996), 429–442, studied the asymptotic behavior of the mean value of Dedekind sums, and H.Rademacher and E.Grosswald, Dedekind Sums, The Carus Mathematical Monographs No. 16, The Mathematical Association of America, Washington, D.C., 1972, studied the related properties. In this paper, we use the algebraic method to study the computational problem of one kind of mean value involving the classical Dedekind sum and the quadratic Gauss sum, and give several exact computational formulae for it.     

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.     

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).     

Dedekind和及第一类Chebyshev多项式

本文利用分析方法、Dedekind和及第一类Chebyshev多项式的算术性质,研究了一类关于Dedekind和及第一类Chebyshev多项式混合均值的渐近估计问题,并得到了一个较强的渐近公式.     

Dedekind和的一个性质

Ｄｅｄｅｋｉｎｄ和的Ｋｎｏｐｐ等式是与Ｈｅｃｋｅ算子有关的一个算术性质，本文不借助ｅｔａ－函数的概念，给予Ｋｎｏｐｐ等式的一个简短的初等证明，同时把Ｋｎｏｐｐ等式拓广到广义Ｄｅｄｅｋｉｎｄ和中。     

On formulas for Dedekind sums and the number of lattice points in tetrahedra

This paper explores a simple yet powerful relationship between the problem of counting lattice points and the computation of Dedekind sums. We begin by constructing and proving a sharp upper estimate for the number of lattice points in tetrahedra with some irrational coordinates for the vertices. Besides providing a sharper estimate, this upper bound (Theorem 1.1) becomes an equality (i.e. gives the exact number of lattice points) in a tetrahedron where the lengths of the edges divide each other. This equality condition can then be applied to the explicit computation of the classical Dedekind sums, a topic that is the central focus in the second half of our paper. In this half of the paper, we come up with a number of interesting results related to Dedekind sums, based on our upper estimate (Theorem 1.1). Among these findings, Theorem 1.9 and Theorem 1.10 deserve special attention, for they successfully generalize two of Apostol's formulas in [T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1997], and also directly imply the famous Reciprocity Law of Dedekind sums.     

关于Fermat素数与L-函数的加权均值

本文研究了一类以Fermat素数为模的Dirichlet L-函数加权均值的计算问题. 利用初等方法以及Dirichlet 和的性质, 获得了一个有趣的计算公式.     

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.     

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.     

Some new identities involving Dedekind sums and the Ramanujan sum

The main purpose of this paper is to introduce new sums that are analogous to Dedekind sums. Using analysis and properties of Dirichlet \(L\) -functions, we study mean values for these new sums, and give a sharper mean value formula for it.     

On generalized Dedekind sums

In this paper we give a simple proof for the reciprocity formula for the generalized Dedekind sums and derive an explicit expression for these sums.     

Cotangent sums,a further generalization of Dedekind sums

In this article a simple proof for a reciprocity formula for sums of cotangent functions is presented. If the second arguments are zero, these sums are four times the original Dedekind sums. Subsequently, explicit expressions are derived. Finally, it is shown that the Berndt sums are special cases of these sums.