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

2025/07/15 by Brown, Steven
Mathematics · #11A05 #11F20 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical Inequalities and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2507.11666

openalex publication_date 2025/07/15 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

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 S_r(n,m)=∑_k=1^n-1\lfloor (km)/(n)\rfloor^r While a formula for S₁ is well known, we provide closed-form formulas for S₂ and S₃ 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 S_r showing its dependency on generalized Dedekind sums.

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