On a family of sums of powers of the floor function and their links with generalized Dedekind sums

TL;DR

This paper derives closed-form formulas and reciprocity laws for sums involving powers of the floor function, linking them to generalized Dedekind sums and expanding understanding of their mathematical properties.

Contribution

It provides new closed-form formulas for sums of powers of the floor function and establishes their connection with generalized Dedekind sums, including reciprocity laws.

Findings

Closed-form formulas for  and  sums

Reciprocity laws for these sums

General formula linking sums to generalized Dedekind sums

Abstract

In this paper we are concerned with a family of sums involving the floor function. With $r$ a non negative integer and $n$ and $m$ positive integers we consider the sums \begin{equation*}\mathbf{S}_{r}\left(n,m\right)=\sum_{k=1}^{n-1}{\left\lfloor \frac{km}{n}\right\rfloor}^r\end{equation*} While a formula for $\mathbf{S}_1$ is well known, we provide closed-form formulas for $\mathbf{S}_2$ and $\mathbf{S}_3$ as well as the reciprocity laws they satisfy. Additionally, one can find a closed-form formula for the classical Dedekind sum using the Euclidean algorithm. Finally, we provide a general formula for $\mathbf{S}_r$ showing its dependency on generalized Dedekind sums.