On a family of sums of powers of the floor function and their links with generalized Dedekind sums
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911058874073088 |
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| author | Brown, Steven |
| author_facet | Brown, Steven |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_11666 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a family of sums of powers of the floor function and their links with generalized Dedekind sums Brown, Steven Number Theory 11F20, 11A05 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. |
| title | On a family of sums of powers of the floor function and their links with generalized Dedekind sums |
| topic | Number Theory 11F20, 11A05 |
| url | https://arxiv.org/abs/2507.11666 |