Generalized Dedekind Sums Research Articles

Fast computation of generalized dedekind sums

We construct an algorithm that reduces the complexity for computing generalized Dedekind sums from exponential to polynomial time. We do so by using an efficient word rewriting process in group theory.

On the Mean Value of the Generalized Dedekind Sum and Certain Generalized Hardy Sums Weighted by the Kloosterman Sum

On the Mean Value of the Generalized Dedekind Sum and Certain Generalized Hardy Sums Weighted by the Kloosterman Sum

On the mean value of the generalized Dedekind sum and certain generalized Hardy sums weighted by the Kloosterman sum

UDC 511 We study a hybrid mean-value problem related to the generalized Dedekind sum, certain generalized Hardy sums, and Kloosterman sum and obtain several meaningful conclusions by means of the analytic method and the properties of the character sum and the Gauss sum.

Algebraic properties of the values of newform Dedekind sums

We study the image of a generalized Dedekind sum relating to the weight zero Eisenstein series Eχ1,χ2. We show that the image is a lattice of full rank inside a number field determined by the characters χ1 and χ2. We also give a generalization of Knopp's identity for the classical Dedekind sum.

Eisenstein cohomology classes for GL N over imaginary quadratic fields

Abstract We study the arithmetic of degree N - 1 {N-1} Eisenstein cohomology classes for the locally symmetric spaces attached to GL N {\mathrm{GL}_{N}} over an imaginary quadratic field k. Under natural conditions we evaluate these classes on ( N - 1 ) {(N-1)} -cycles associated to degree N extensions L / k {L/k} as linear combinations of generalized Dedekind sums. As a consequence we prove a remarkable conjecture of Sczech and Colmez expressing critical values of L-functions attached to Hecke characters of L as polynomials in Kronecker–Eisenstein series evaluated at torsion points on elliptic curves with complex multiplication by k. We recover in particular the algebraicity of these critical values.

On the generalized Cochrane sum with Dirichlet characters

<abstract><p>In this paper, we defined a new generalized Cochrane sum with Dirichlet characters, and gave the upper bound of the generalized Cochrane sum with Dirichlet characters. Moreover, we studied the asymptotic estimation problem of the mean value of the generalized Cochrane sum with Dirichlet characters and obtained a sharp asymptotic formula for it. By using this asymptotic formula, we also gave the mean value of the generalized Dedekind sum.</p></abstract>

Poly-Dedekind sums associated with poly-Bernoulli functions

Apostol considered generalized Dedekind sums by replacing the first Bernoulli function appearing in Dedekind sums by any Bernoulli functions and derived a reciprocity relation for them. Recently, poly-Dedekind sums were introduced by replacing the first Bernoulli function appearing in Dedekind sums by any type 2 poly-Bernoulli functions of arbitrary indices and were shown to satisfy a reciprocity relation. In this paper, we consider other poly-Dedekind sums that are obtained by replacing the first Bernoulli function appearing in Dedekind sums by any poly-Bernoulli functions of arbitrary indices. We derive a reciprocity relation for these poly-Dedekind sums.

Identities on poly-Dedekind sums

Dedekind sums occur in the transformation behavior of the logarithm of the Dedekind eta-function under substitutions from the modular group. In 1892, Dedekind showed a reciprocity relation for the Dedekind sums. Apostol generalized Dedekind sums by replacing the first Bernoulli function appearing in them by any Bernoulli functions and derived a reciprocity relation for the generalized Dedekind sums. In this paper, we consider the poly-Dedekind sums obtained from the Dedekind sums by replacing the first Bernoulli function by any type 2 poly-Bernoulli functions of arbitrary indices and prove a reciprocity relation for the poly-Dedekind sums.

Dedekind sums arising from newform Eisenstein series

For primitive nontrivial Dirichlet characters [Formula: see text] and [Formula: see text], we study the weight zero newform Eisenstein series [Formula: see text] at [Formula: see text]. The holomorphic part of this function has a transformation rule that we express in finite terms as a generalized Dedekind sum. This gives rise to the explicit construction (in finite terms) of elements of [Formula: see text]. We also give a short proof of the reciprocity formula for this Dedekind sum.

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 $\Gamma_0(N)^+$.

An average of generalized Dedekind sums

We study a generalization of the classical Dedekind sum that incorporates two Dirichlet characters and develop properties that generalize those of the classical Dedekind sum. By calculating the Fourier transform of this generalized Dedekind sum, we obtain an explicit formula for its second moment. Finally, we derive upper and lower bounds for the second moment with nearly identical orders of magnitude.

Arithmetic of generalized Dedekind sums and their modularity

Abstract Dedekind sums were introduced by Dedekind to study the transformation properties of Dedekind η function under the action of SL2(ℤ). In this paper, we study properties of generalized Dedekind sums s i,j (p, q). We prove an asymptotic expansion of a function on ℚ defined in terms of generalized Dedekind sums by using its modular property. We also prove an equidistribution property of generalized Dedekind sums.

Generalized Dedekind sums and the transformation formula in finite fields

We use periodic functions to introduce the generalized Dedekind sums in finite fields to describe the transformation formula for a family of certain functions.

On the integral of products of higher-order Bernoulli and Euler polynomials

In this paper, we derive a formula on the integral of products of the higher-order Euler polynomials. By the same way, similar relations are obtained for $l$ higher-order Bernoulli polynomials and $r$ higher-order Euler polynomials. Moreover, we establish the connection between the results and the generalized Dedekind sums and Hardy--Berndt sums. Finally, the Laplace transform of Euler polynomials is given.

New Trigonometric Identities and Reciprocity Laws of Generalized Dedekind Sums

In this paper, we prove new trigonometric identities, which are product-to-sum type formulas for the higher derivatives of the cotangent and cosecant functions. Furthermore, from specializations of our formulas, we derive various known and new reciprocity laws of generalized Dedekind sums.

Generalized Dedekind sums and transformation formulae in function fields

Our previous research involving the use of the Dedekind sum defined by Bayad and the author enabled us to establish the transformation formula for a certain series in function fields. In this work we use periodic functions to introduce the generalized Dedekind sums in function fields to describe transformation formulae for a family of certain series.

On the theorem of Davenport and generalized Dedekind sums

A symmetrized lattice of 2n points in terms of an irrational real number α is considered in the unit square, as in the theorem of Davenport. If α is a quadratic irrational, the square of the L2 discrepancy is found to be c(α)log⁡n+O(log⁡log⁡n) for a computable positive constant c(α). For the golden ratio φ, the value c(φ)log⁡n yields the smallest L2 discrepancy of any sequence of explicitly constructed finite point sets in the unit square. If the partial quotients ak of α grow at most polynomially fast, the L2 discrepancy is found in terms of ak up to an explicitly bounded error term. It is also shown that certain generalized Dedekind sums can be approximated using the same methods. For a special generalized Dedekind sum with arguments a, b an asymptotic formula in terms of the partial quotients of ab is proved.

The p-adic analytic Dedekind sums

In this paper, using Cohen's and Tangedal and Young's theory on the p-adic Hurwitz zeta functions, we construct the analytic Dedekind sums on the p-adic complex plane Cp. We show that these Dedekind sums interpolate Carlitz's higher order Dedekind sums p-adically. From Apostol's reciprocity law for the generalized Dedekind sums, we also prove a reciprocity relation for the special values of these p-adic Dedekind sums. Finally, the parallel results for the analytic Dedekind sums on the p-adic complex plane associated with Euler polynomials have also been given.

Higher Hickerson formula

In [11], Hickerson made an explicit formula for Dedekind sums s(p,q) in terms of the continued fraction of p/q. We develop analogous formula for generalized Dedekind sums si,j(p,q) defined in association with the xiyj-coefficient of the Todd power series of the lattice cone in R2 generated by (1,0) and (p,q). The formula generalizes Hickerson's original one and reduces to Hickerson's for i=j=1. In the formula, generalized Dedekind sums are divided into two parts: the integral sijI(p,q) and the fractional sijR(p,q). We apply the formula to Siegel's formula for partial zeta values at a negative integer and obtain a new expression which involves only sijI(p,q) the integral part of generalized Dedekind sums. This formula directly generalizes Meyer's formula for the special value at 0. Using our formula, we present the table of the partial zeta value at s=−1 and −2 in more explicit form. Finally, we present another application on the equidistribution property of the fractional parts of the graph (pq,Ri+jqi+j−2sij(p,q)) for a certain integer Ri+j depending on i+j.

EXPLICIT FORMULA FOR COEFFICIENTS OF TODD SERIES OF LATTICE CONES

Abstract. Todd series are associated to maximal non-degenerate latticecones. The coefficients of Todd series of a particular class of lattice conesare closely related to generalized Dedekind sums of higher dimension. Wegeneralize this construction and obtain an explicit formula for coefficientsof the Todd series. It turns out that every maximal non-degenerate latticecone, hence the associated Todd series can be obtained in this way. 1. IntroductionIn [2, 3], we have defined the following generalized Dedekind sums of higherdimension and considered their properties including integrality, equidistribu-tion and reciprocity: for q ∈ Z >0 , (a 1 ,...,a n ) ∈ Z n and (i 1 ,...,i n ) ∈ Z n≥0 ,(1)X (k 1 ,...,k n ) Be i 1 k 1 qBe i 2 k 2 ···Be i n k n ,where the summation is taken over the set of n-tuples (k 1 ,...,k n ) of non-negative integers less that q such that a 1 k 1 + ··· + a n k n ≡ 0 mod q. TheBe k (x) denotes the k-th periodic Bernoulli function, which is equal to the k-thBernoulli polynomial B