🔢 mathematics

On the calculation of an integral introduced by Hubert Delange

This paper investigates Hubert Delange's previously unsolved integral concerning the distribution of reduced fractions, establishing general properties that reduce the problem to odd square-free integers and providing a specific formula for the case $q=2$ that expresses the integral in terms of fourfold Möbius sums and generalized Dedekind sums.

Original authors: Steven Brown

Published 2026-10-14✓ Author reviewed ⓘ

📖 4 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 by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of number theory, mathematicians often study the hidden patterns within whole numbers. One of the most fundamental concepts in this field is the idea of counting how many numbers share a specific relationship with a given integer. For any whole number greater than one, there are other numbers smaller than it that do not share any common factors with it, except for the number one itself. These special numbers are called totatives. If you were to list all the totatives for a large number, they would not be scattered randomly; they would form a specific, structured sequence. The question of how these numbers are distributed has fascinated scholars for over a century, leading to deep investigations into whether they spread out evenly or clump together in unexpected ways.

To measure this distribution, researchers look at the gaps between these totatives and where they "should" be if they were perfectly spaced out. Imagine lining up a set of books on a shelf; if they were perfectly evenly spaced, the distance between each book would be identical. In the world of numbers, the totatives rarely line up with such perfect precision. Mathematicians have developed a way to calculate the total error in this spacing by squaring the differences and adding them up. This calculation results in a specific value, or integral, that tells us how far the actual distribution deviates from a perfect, uniform spread. For decades, this calculation was only fully understood for a simple case where the error was measured in a basic way. However, when mathematicians tried to measure the error with greater sensitivity, looking at more complex variations of the spacing, the problem became incredibly difficult, and the formula remained a mystery.

This paper tackles that long-standing mystery by taking a fresh look at the mathematical tools used to solve it. The author, Steven Brown, begins by acknowledging a famous observation made by Hubert Delange, a mathematician who had successfully solved the simple version of the problem but admitted he could not find a solution for the more complex versions. Brown's work does not immediately provide a single, neat formula for every possible case, but it significantly narrows the field of search. He demonstrates that the complex problem can be broken down into smaller, more manageable pieces. Specifically, he proves that the behavior of these numbers depends only on the unique prime factors of the number being studied, ignoring any repeated factors. This means that to understand the pattern for a massive, complicated number, one only needs to understand the pattern for a much simpler number made of distinct prime factors.

The paper then focuses on a specific, slightly more complex version of the problem, where the error is measured with a higher degree of sensitivity. For this specific case, the author succeeds in reducing the entire calculation to a single, difficult sum. This sum involves a specific type of counting related to how numbers wrap around in cycles, a concept known in mathematics as modular arithmetic. While the author cannot yet provide a simple, closed-form formula for this final sum, he has successfully translated the original, intimidating problem into a new form that is much easier to study. He shows that the answer depends on a four-part sum involving the Möbius function, a tool used to filter numbers based on their prime factors, and a type of sum related to generalized Dedekind sums, which are known for their deep connections to geometry and number theory.

The findings are precise and rigorous. The author proves that for certain types of numbers, the complex error calculation is actually zero, a result that confirms earlier suspicions about the symmetry of these distributions. For other cases, he provides a detailed formula that links the error to the number of prime factors and the total count of totatives. The work essentially maps out the terrain, showing that the difficult part of the journey is now isolated to a specific, well-defined mathematical object. The paper concludes by suggesting that future researchers can now focus their efforts on understanding this specific sum, perhaps using different mathematical techniques like those involving wave patterns or direct counting methods. The mystery is not fully solved, but the path forward has been cleared, turning a vague, overwhelming problem into a concrete target for further discovery.

Technical Summary: On the calculation of an integral introduced by Hubert Delange

Problem Statement
This paper addresses the evaluation of a family of integrals introduced by Hubert Delange in his study of the asymptotic distribution of reduced fractions (totatives) relative to equidistributed numbers. The integral is defined as:
$\int_0^1 \left( \frac{N(n, t)}{\phi(n)} - t \right)^{2q} dt$
where $n > 1$ and $q > 0$ are integers, $\phi$ is the Euler totient function, and $N(n, t)$ counts the number of reduced fractions with denominator $n$ in the interval $]0, t]$ . While Delange successfully evaluated this integral for $q=1$ (yielding a formula involving the number of distinct prime divisors $\omega(n)$ ), he noted that he was unable to evaluate it for $q > 1$ . The paper aims to provide closed-form formulas for this integral for fixed $q > 1$ and arbitrary $n$ .

Methodology
The author reformulates the problem by introducing the function $\nu_n(x)$ , related to the fractional part of $x/d$ summed over divisors $d|n$ with Möbius weights. The integral is transformed into an equivalent form, $H_k(n)$ , defined as:
$H_k(n) := \int_0^1 (-\nu_n(nt))^k dt$
where $k=2q$ . The methodology proceeds through several analytical steps:

Reduction to Franel Integrals: The author expresses $H_k(n)$ as a $k$ -fold Möbius sum of Franel integrals, $F(a_1, \dots, a_k)$ , which are integrals of products of periodic Bernoulli polynomials.
Symmetry and Reduction: It is proven that $H_k(n) = 0$ for odd $k$ . Furthermore, the value of $H_k(n)$ depends only on the square-free part of $n$ (denoted $\text{rad}(n)$ ), and for odd $m$ , $H_k(2m) = H_k(m)$ . This reduces the general problem to calculating $H_k(m)$ where $m$ is an odd, square-free integer.
Connection to Totative Sums: Theorem 2.1 establishes a direct relationship between $H_k(n)$ and the sums of powers of $\nu_n(a)$ over the set of totatives $t(n)$ . Specifically, $H_k(n)$ is expressed in terms of $V_{k+1}(n) = \sum_{a \in t(n)} \nu_n(a)^{k+1}$ .
Evaluation via Modular Sums: For the specific case of $k=4$ (corresponding to $q=2$ ), the author utilizes a direct approach to Franel integrals. This involves subdividing the integration interval based on the least common multiple of the parameters and expressing the integral in terms of integer-valued modular sums $W_N(E)$ and $X_N(A)$ . These sums are linked to generalized Dedekind sums.

Key Contributions and Results

General Properties: The paper proves that $H_k(n)$ vanishes for odd $k$ and is invariant under the removal of square factors from $n$ .
Prime Power Case: A closed-form formula is derived for $H_k(p^\alpha)$ where $p$ is prime, utilizing Faulhaber's formula for sums of powers.
Reduction for $q=2$ ( $k=4$ ): The problem of evaluating $H_4(n)$ is reduced to the calculation of the sum $V_5(n)$ . While the paper notes that formulas for $V_i(n)$ are known for $i \le 3$ , it explicitly states that no formula has been found for $V_4(n)$ . However, through a system of linear equations derived from Proposition 3.1, the author demonstrates that $V_5(n)$ depends on $V_4(n)$ . Consequently, the evaluation of $H_4(m)$ (where $m$ is square-free) is reduced to determining the value of the fourfold Möbius sum $W_4(m)$ , which encapsulates the unknown $V_4(n)$ component.
Explicit Formula for $H_4(m)$ : Theorem 3.1 provides the explicit decomposition:
$H_4(m) = W_4(m) + \frac{1}{60m} \left( \frac{\phi(m)}{m} \right)^3 (2\phi(m) - 5m^2\omega(m))$
where $W_4(m)$ is defined as a fourfold Möbius sum of Franel integrals of order 4.
Franel Integral Formulas: The paper provides closed-form formulas for the auxiliary sums $X_N(A)$ (which determine the Franel integrals) for sets $A$ with cardinality $|A| \le 3$ . Consequently, it offers a complete calculation for Franel integrals of order 2. However, for order 4 (where $|A|=4$ ), the author notes that no simple closed-form formula for the sum $X_N(A)$ (or the integral itself) has been found; the result for order 4 isolates the non-elementary term $W_N(E)$ .

Significance and Scope
The paper does not claim to have solved the general problem for all $q > 1$ . Instead, it significantly narrows the scope of the unsolved problem. The author states that the calculation of the integral for $q=2$ is now equivalent to evaluating the fourfold Möbius sum $W_4(m)$ (which is intrinsically linked to the unknown $V_4(m)$ ), involving integer-valued modular sums related to generalized Dedekind sums.

The paper concludes modestly, noting that while a general closed form for Franel integrals of order 4 (and thus $W_4(m)$ ) is not found, the problem has been reduced to a specific arithmetic calculation. The author suggests future work could involve applying Parseval's theorem to the periodic function $\nu_n(nt)^2$ , direct calculation of $V_4(m)$ using Spence's ideas, or analyzing the reciprocity laws of the sums $W_N$ . The primary contribution is the rigorous reduction of Delange's open problem to the evaluation of these specific modular sums.

More like this