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

This paper presents a new proof of Spence's 1963 formula for the sum of products of coprime integers less than $n$ and their indices, utilizing the reciprocity law for Dedekind sums as an alternative to previous methods involving Nagell's totient function or Fourier analysis.

Original authors: Steven Brown

Published 2026-01-30

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Steven Brown

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A New Way to Count

Imagine you have a large number, let's call it $n$ . Now, imagine a club of "special numbers" that are smaller than $n$ and have a very specific relationship with it: they don't share any common factors with $n$ (mathematicians call these "coprime").

If you list these special numbers in order from smallest to largest, you get a sequence: $a_1, a_2, a_3, \dots$ .

In 1963, a mathematician named Edward Spence discovered a magical formula. This formula allows you to calculate the sum of these numbers, but with a twist: you multiply each number by its position in the line.

Take the first number ( $a_1$ ) and multiply it by 1.
Take the second number ( $a_2$ ) and multiply it by 2.
Do this for the whole list and add them all up.

Spence found that this total sum isn't random; it follows a precise pattern based on the size of $n$ and how many prime factors it has.

The Problem: Spence proved this in 1963, and another mathematician (Lucien Van Hamme) proved it again in 1971 using a different method (Fourier analysis, which is like breaking a sound wave into its notes).

The Goal of This Paper: Steven Brown wants to prove this same formula a third time, but using a completely different tool: Dedekind sums and their "Reciprocity Law."

The Tools of the Trade

To understand Brown's proof, we need to understand the three main tools he uses, which he treats like ingredients in a recipe.

1. The "Filter" (The Coprime Club)

Brown starts by looking at the list of special numbers ( $a_1$ to $a_{\phi(n)}$ ). Instead of trying to force them into order immediately, he uses a mathematical "filter" (called the function $\theta_n$ ).

Analogy: Imagine you have a bucket of mixed marbles. You want to count only the red ones. Instead of sorting them one by one, you pour the bucket through a sieve that only lets red marbles through. This function $\theta_n$ acts as that sieve, counting how many valid numbers exist up to a certain point without worrying about their specific order yet.

2. The "Fractional Part" (The Leftover Crumbs)

In math, when you divide a number, you get a whole number and a "fractional part" (the leftovers). For example, $7 \div 3 = 2$ with a leftover of $1/3$ .
Brown focuses heavily on these "leftovers." He uses a special notation, $((x))$ , which measures how far a number is from the nearest half-integer.

Analogy: Think of a clock. The "fractional part" is where the hand is between the numbers. Brown is interested in the symmetry of these positions. He uses a

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

Problem Statement
The paper addresses the evaluation of the weighted sum of integers coprime to a given integer $n$ . Specifically, let $n > 1$ be an integer, and let $0 < a_1 < a_2 < \dots < a_{\phi(n)} < n$ be the positive integers less than $n$ that are coprime to $n$ , arranged in ascending order. The objective is to compute the sum:
$\sum_{j=1}^{\phi(n)} j a_j$
This formula was originally established by Edward Spence in 1963 and later reproven by Lucien Van Hamme in 1971 using Fourier analysis. The paper aims to provide a third, distinct proof utilizing the theory of Dedekind sums.

Methodology
The author, Steven Brown, employs a method rooted in analytic number theory, specifically leveraging the properties of Dedekind sums and the Möbius inversion formula. The proof proceeds through the following logical steps:

Reformulation via Counting Functions:
Instead of summing over the ordered indices $j$ , the author rewrites the target sum using the function $\theta_n(x)$ , which counts the number of integers coprime to $n$ less than or equal to $x$ . This allows the sum to be expressed over the set $U(n)$ of coprime integers without explicit reference to their ordering:
$\sum_{j=1}^{\phi(n)} j a_j = \sum_{a \in U(n)} \theta_n(a) a$
Decomposition using Fractional Parts:
Using the identity $\theta_n(x) + \nu_n(x) = x\phi(n)/n$ , where $\nu_n(x)$ involves fractional parts $\{x/d\}$ , the problem is decomposed. The term involving $\nu_n(a)$ is isolated. The author expresses the fractional part $\{a/d\}$ in terms of the sawtooth function $((x)) = \{x\} - 1/2$ (for non-integers) and utilizes the property that $\sum_{a \in U(n)} ((a/d)) = 0$ .
Introduction of Dedekind Sums:
The remaining term involves a double sum over divisors $d_1, d_2$ of $n$ containing products of sawtooth functions. This structure is identified as a linear combination of classical Dedekind sums, denoted $s(b, a)$ . The expression is transformed into:
$S(n) = n \sum_{d_1|n} \sum_{d_2|n} \mu(d_1)\mu(d_2) s\left(\frac{n}{d_1}, \frac{n}{d_2}\right)$
Application of Reciprocity Law:
The core of the proof utilizes the symmetry of the sum and the Dedekind reciprocity law. By grouping terms $s(b, a)$ and $s(a, b)$ , the author applies the reciprocity formula:
$$ s

The Big Picture: A New Way to Count

The Tools of the Trade

1. The "Filter" (The Coprime Club)

2. The "Fractional Part" (The Leftover Crumbs)

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

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