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

Abstract

In 1963, Edward Spence published a proof of the following With $\phi$ being Euler 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.