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Reciprocity formulas for certain generalized Hardy-Berndt sums

Yuan He

TL;DR

This work extends Hardy–Berndt reciprocity to a broad class of generalized sums $S_{m,n}^{(5)}$ built from the quasi-periodic Euler functions $\overline{E}_m$ and Bernoulli functions $\overline{B}_n$ by employing Fourier-series methods and the periodic zeta/Lerch zeta frameworks. The authors derive new reciprocity formulas, including non-coprime and multi-parameter extensions, which subsume Hardy’s original reciprocity as a special case. The results are presented via two main theorems, with detailed auxiliary lemmas enabling explicit evaluations of Fourier coefficients and sums over diophantine solutions. Collectively, the paper broadens the landscape of reciprocity relations for generalized Hardy-Berndt sums and connects them to classical zeta-function techniques, offering new identities and corollaries such as refined forms of $s_{5}(a,b)$ in terms of GCDs and Euler/Bernoulli constructs.

Abstract

In this paper, we establish some reciprocity formulas for certain generalized Hardy-Berndt sums by using the Fourier series technique and some properties of the periodic zeta function and the Lerch zeta function. It turns out that one of Hardy's reciprocity theorems is deduced as a special case.

Reciprocity formulas for certain generalized Hardy-Berndt sums

TL;DR

This work extends Hardy–Berndt reciprocity to a broad class of generalized sums built from the quasi-periodic Euler functions and Bernoulli functions by employing Fourier-series methods and the periodic zeta/Lerch zeta frameworks. The authors derive new reciprocity formulas, including non-coprime and multi-parameter extensions, which subsume Hardy’s original reciprocity as a special case. The results are presented via two main theorems, with detailed auxiliary lemmas enabling explicit evaluations of Fourier coefficients and sums over diophantine solutions. Collectively, the paper broadens the landscape of reciprocity relations for generalized Hardy-Berndt sums and connects them to classical zeta-function techniques, offering new identities and corollaries such as refined forms of in terms of GCDs and Euler/Bernoulli constructs.

Abstract

In this paper, we establish some reciprocity formulas for certain generalized Hardy-Berndt sums by using the Fourier series technique and some properties of the periodic zeta function and the Lerch zeta function. It turns out that one of Hardy's reciprocity theorems is deduced as a special case.
Paper Structure (3 sections, 11 theorems, 68 equations)

This paper contains 3 sections, 11 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Some auxiliary results
  3. The proofs of Theorems \ref{['thm1.1']} and \ref{['thm1.3']}

Key Result

Theorem 1.1

Let $m,n\in\mathbb{N}_{0}$, $a,b\in\mathbb{Z}$ with $a\equiv b\equiv 1$ (mod $2$). Then, for $x,y,z\in\mathbb{R}$, where $\mathrm{sgn}(x)$ indicates the sign of $x$ given for $x\in\mathbb{R}\setminus\{0\}$ by $\mathrm{sgn}(x)=x/|x|$, $\delta_{l,k}$ is the Kronecker delta function given for $l,k\in\mathbb{N}_{0}$ by $\delta_{l,k}=1$ or $0$ according to $l=k$ or $l\not=k$, $\delta_{\mathbb{Z}}(x)=1

Theorems & Definitions (18)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 8 more