A proof of Spence’s formula using the reciprocity law for Dedekind sums

Steven Brown
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 2, Pages 335–341
DOI: 10.7546/nntdm.2026.32.2.335-341
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Steven Brown

48 rue Pottier, 78150 Le Chesnay Rocquencourt, France

Abstract

In 1963, Edward Spence published a proof of the following:

With \phi being Euler’s totient function, if n>1 is an integer, and if

    \begin{equation*} 0<a_1<\cdots<a_{\phi(n)}<n, \end{equation*}

are the positive integers less than n, coprime with n, then

    \begin{equation*} \sum_{j=1}^{\phi(n)}ja_j = \frac{\phi(n)}{24}\left(8n\phi(n)+6n+2\phi(m)(-1)^{\omega(m)}-2^{\omega(m)}\right), \end{equation*}

where m is the square-free part of n, and \omega(m) is the number of prime factors of m.

Spence’s proof relies on an ingenious observation considering Nagell’s totient function.
Later in 1971, Lucien Van Hamme provided an alternative proof of the result using Fourier analysis and previous work from Hubert Delange in 1968. In this paper, I propose another proof of the formula using the reciprocity law for Dedekind sums. If the formula is of interest on its own, it also plays a role in the analysis of the distribution of the a_j as suggested by the work from Hubert Delange.

Keywords

  • Spence formula
  • Dedekind sums
  • Arithmetical functions

2020 Mathematics Subject Classification

  • 11A99
  • 11F20
  • 11A25

References

  1. Apostol, T. M. (1976). Introduction to Analytic Number Theory. Springer-Verlag, New York.
  2. Delange, H. (1968). Sur la distribution des fractions irréductibles de dénominateur n ou de dénominateur au plus égalà x. Hommage au Professeur Lucien Godeaux, 75–89.
  3. Dickson, L. E. (1919). History of the Theory of Numbers, Vol. I. Chelsea, New York.
  4. Rademacher, H., & Grosswald, E. (1972). Dedekind Sums. The Carus Mathematical Monographs, Volume 16. American Mathematical Society, Washington, D.C.
  5. Sándor, J., & Crstici, B. (2004). Handbook of Number Theory, II. Springer Science & Business Media.
  6. Spence, E. (1963). Formulae for sums involving a reduced set of residues modulo n. Proceedings of the Edinburgh Mathematical Society, 13(4), 347–349.
  7. Van Hamme, L. (1971). Sur une généralisation de l’indicateur d’Euler. Bulletins de l’Académie Royale de Belgique, 57(1), 805–817.

Manuscript history

  • Received: 28 February 2026
  • Revised: 24 May 2026
  • Accepted: 30 May 2026
  • Online First: 3 June 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).

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Cite this paper

Brown, S. (2026). A proof of Spence’s formula using the reciprocity law for Dedekind sums. Notes on Number Theory and Discrete Mathematics, 32(2), 335-341, DOI: 10.7546/nntdm.2026.32.2.335-341.

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