Higher-order character Dedekind sum

Mümün Can

doi:10.3906/mat-1911-93

Abstract

In this paper, we are interested in higher-order character Dedekind sum ck∑−1 v=0 χ1 v Bp,χ2 a v + z c + x Bq b v + z ck + y , a, b, c ∈ N and x, y, z ∈ R, where χ1 and χ2 are primitive characters of modulus k, Bp x and Bp,χ2 x are Bernoulli and generalized Bernoulli functions, respectively. We employ the Fourier series technique to demonstrate reciprocity formulas for this sum. Derived formulas are analogues of Mikolas’ reciprocity formula. Moreover, we offer Petersson–Knopp type identities for this sum.

Highlights

Let Bn (x) denote the n th Bernoulli function defined by { Bn (x) =Bn (x − [x]), if n= 1 or x ∈/ Z, 0, if n = 1 and x ∈ Z, where [x] denotes the largest integer ≤ x and Bn(x) is the n th Bernoulli polynomial defined by text et − 1 = ∑ ∞ Bn(x) tn. n! n=0The classical Dedekind sum s(a, c), arising in the theory of Dedekind eta-function, is defined by∑ c−1 ( v ) s(a, c) = B1 c B1 c, a, c ∈ Z, c > 0
This sum contains generalized Dedekind sums previously defined by Carlitz [17]
Reciprocity formulas of Hall et al [25], Bayad and Raouj [5] and Beck and Chavez [6] are in terms of the generating functions

Summary

Introduction

Due to Hall et al [25], is the higher-order Dedekind sum (or generalized Dedekind–Rademacher sum). This sum contains generalized Dedekind sums previously defined by Carlitz [17] (see [8, 40]). Reciprocity formulas of Hall et al [25], Bayad and Raouj [5] and Beck and Chavez [6] are in terms of the generating functions. Where (a, c) = 1 (for several proofs see [38]), whereas the reciprocity formula for the sum s(a, c; χ) is [7, Theorem 4]. We are interested in character extension of (1.1), i.e., higher-order character Dedekind sum c∑ k−1

Bq b ck

We now utilize the partial fractions