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Some reciprocity formulas for generalized Dedekind-Rademacher sums

Yuan He, Yong-Guo Shi

Abstract

In this paper, we study the generalized Dedekind-Rademacher sums considered by Hall, Wilson and Zagier. We establish a formula for the products of two Bernoulli functions. The proof relies on Parseval's formula, Hurwitz's formula, and Lerch's functional equation. The result leads to reciprocity formulas for some generalizations of the classical Dedekind sums. In particular, it is shown that Carlitz's, Berndt's, Hall and Wilson's reciprocity theorems are deduced as special cases.

Some reciprocity formulas for generalized Dedekind-Rademacher sums

Abstract

In this paper, we study the generalized Dedekind-Rademacher sums considered by Hall, Wilson and Zagier. We establish a formula for the products of two Bernoulli functions. The proof relies on Parseval's formula, Hurwitz's formula, and Lerch's functional equation. The result leads to reciprocity formulas for some generalizations of the classical Dedekind sums. In particular, it is shown that Carlitz's, Berndt's, Hall and Wilson's reciprocity theorems are deduced as special cases.
Paper Structure (4 sections, 17 theorems, 113 equations)

This paper contains 4 sections, 17 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Some auxiliary results
  3. Formulas of products of Bernoulli functions
  4. More general reciprocity formulas

Key Result

Lemma 2.1

(Parseval's formula) Suppose that $F(\theta)$ and $G(\theta)$ are two Riemann integrable, complex-valued functions on $\mathbb{R}$ of period $2\pi$ with the Fourier series and Then where the horizontal bars indicate complex conjugation.

Theorems & Definitions (34)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 24 more