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

Steven Brown
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 1, Pages 76–87
DOI: 10.7546/nntdm.2026.32.1.76-87
Full paper (PDF, 265 Kb)

Details

Authors and affiliations

Steven Brown
48 rue Pottier, 78150 Le Chesnay Rocquencourt, France

Abstract

In this paper we are concerned with a family of sums involving the floor function. With r a nonnegative integer and n and m positive integers we consider the sums

    \[\mathbf{S}_{r}(n,m):=\sum_{k=1}^{n-1}{\left\lfloor \frac{km}{n}}\right\rfloor ^r.\]

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.

Keywords

  • Sum of powers of the floor function
  • Dedekind sums
  • Faulhaber sums
  • Reciprocity laws
  • Euclidean algorithm

2020 Mathematics Subject Classification

  • 11F20
  • 11A05

References

  1. Damphousse, P. (2000). Opuscules. Découvrir l’Arithmétique. Ellipses.
  2. Graham, R. L., Knuth, D. E., & Patashnik, O. (1994). Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley Professional.
  3. Polezzi, M. (1997). A geometrical method for finding an explicit formula for the greatest common divisor. The American Mathematical Monthly, 104(5), 445–446.
  4. Rademacher, H., & Grosswald, E. (1972). Dedekind Sums (Vol. 16). American
    Mathematical Society.
  5. Schumacher, R. (2016). An extended version of Faulhaber’s formula. Journal of Integer Sequences, 19(4), Article 16.4.
  6. Zagier, D. (1973). Higher dimensional Dedekind sums. Mathematische Annalen, 202, 149–172.

Manuscript history

  • Received: 3 October 2025
  • Revised: 20 February 2026
  • Accepted: 23 February 2026
  • Online First: 23 February 2026

Copyright information

Ⓒ 2026 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Related papers

Cite this paper

Brown, S. (2026). On a family of sums of powers of the floor function and their links with generalized Dedekind sums. Notes on Number Theory and Discrete Mathematics, 32(1), 76-87, DOI: 10.7546/nntdm.2026.32.1.76-87.

Comments are closed.