# On a recent reciprocity formula for Dedekind sums

## TL;DR

This paper explores a family of reciprocity formulas for Dedekind sums, extending recent results by using the three-term relation rather than L-series connections, revealing deeper structural properties.

## Contribution

It demonstrates that Du and Zhang's reciprocity formula is a special case within a broader class of formulas derived via the three-term relation.

## Key findings

- Generalizes reciprocity formulas for Dedekind sums
- Shows the formulas form a series of related identities
- Uses the three-term relation as the main analytical tool

## Abstract

Let $s(a,b)$ denote the classical Dedekind sum and $S(a,b)=12s(a,b)$. Recently, Du and Zhang proved the following reciprocity formula. If $a$ and $b$ are odd natural numbers, $(a,b)=1$, then $$ S(2a^*,b)+S(2b^*,a)=\frac{a^2+b^2+4}{2ab}-3, $$ where $aa^*\equiv 1\mod b$ and $bb^* \equiv 1 \mod a$. In this paper we show that this formula is a special case of a series of similar reciprocity formulas. Whereas Du and Zhang worked with the connection of Dedekind sums and values of $L$-series, our main tool is the three-term relation for Dedekind sums.