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A statistical investigation of a divisor-sum function

This paper investigates the statistical properties of the divisor-sum function $S_s(n)$ , proving that $S_s(n)/n$ possesses a continuous asymptotic distribution with dense values in $[0, \infty)$ , while establishing mean values and moment bounds using a novel method by Lebowitz-Lockard and Pollack.

Original authors: Ivan Aidun, Lola Thompson

Published 2026-04-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ivan Aidun, Lola Thompson

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

Imagine you have a giant, chaotic library of numbers. For over 2,000 years, mathematicians have been fascinated by one specific rule in this library: the Sum of Proper Divisors.

Think of a number like a social gathering. The "divisors" are all the guests who can divide the host's age evenly. The "proper divisors" are everyone except the host themselves.

If the sum of the guests' ages is less than the host's age, the host is "deficient" (feeling a bit empty).
If the sum is more, the host is "abundant" (feeling rich).
If the sum is exactly equal, the host is "perfect."

For centuries, we've studied these hosts. But in this paper, authors Ivan Aidun and Lola Thompson decided to look at a new, more complex character: a function they call $S_s(n)$ .

The New Character: The "Grand Sum"

If the original rule was just summing the guests at one party, $S_s(n)$ is like summing the guests at every single party that the number $n$ has ever hosted, plus the guests at every party those guests hosted, and so on. It's a massive, nested calculation.

The authors wanted to know: If we look at this giant sum for every number in the universe, what does the pattern look like?

Here are the four main discoveries they made, explained with simple analogies:

1. The "Dense" Crowd (The Values are Everywhere)

The Finding: The values of this new function, when divided by the number itself, are dense in the interval from 0 to infinity.

The Analogy: Imagine you are trying to hit a target with a dart, but the target is the entire number line from 0 to infinity. Usually, mathematical functions are like a dartboard with specific holes; you can only land on certain numbers.
However, the authors proved that for this new function, you can hit any spot on that number line. If you pick a target number (say, 3.14159), you can find a number $n$ where the function's value is almost exactly that. There are no "gaps" in the crowd; the values are packed so tightly that they cover the entire range.

2. The Smooth Curve (The Distribution)

The Finding: The function has a continuous distribution function.

The Analogy: Imagine you are pouring sand through a sieve.

Some mathematical functions are like a sieve with big, jagged holes. The sand falls through in clumps, creating a "jagged" or "singular" pattern.
Other functions are like a smooth waterfall. The sand flows continuously.
The authors proved that this new function behaves like a smooth waterfall. If you plot the probability of the function landing in a certain range, you get a perfectly smooth, unbroken curve. There are no sudden jumps or gaps.

Why is this hard? Usually, to prove a curve is smooth, you use a standard "recipe" (a famous theorem by Erdős and Wintner). But the authors found that the ingredients for this specific function were "spoiled" (mathematically speaking, the standard recipe failed because the function isn't "multiplicative" in the usual way).
The Solution: They used a new, modern tool (developed by Lebowitz-Lockard and Pollack) that acts like a specialized wrench. This tool allowed them to bypass the broken parts of the old recipe and prove the curve is smooth anyway.

3. The Average Behavior (The Mean)

The Finding: They calculated the exact "average" value of this function.

The Analogy: If you asked every person in a stadium to shout out their height, you could calculate the average height. The authors did the same for this mathematical function across all integers. They found a specific, precise number that represents the "typical" behavior of this function. It's not random chaos; it has a steady, predictable center of gravity.

4. The "Extreme" Limits (The Moments)

The Finding: They looked at the "moments" (which measure how wild or extreme the values get) and proved they are under control.

The Analogy: Imagine a rollercoaster.

The "average" is the height of the track.
The "moments" tell you how high the loops go and how fast the drops are.
The authors proved that while the rollercoaster can go very high (the values can get large), it doesn't go infinitely wild. The "extremes" are bounded in a predictable way. They showed that even the wildest swings of this function follow a specific, manageable growth pattern.

Why Does This Matter?

This paper is a bit like a detective story.

The Mystery: We have a new, complicated mathematical object ( $S_s(n)$ ) that doesn't play by the old rules.
The Obstacle: The classic tools of the trade (the "old recipes") break when applied to it.
The Breakthrough: The authors used a brand-new, cutting-edge method to solve the mystery.

They showed that even though this function is messy and complex, it still follows deep, beautiful statistical laws. It's smooth, it covers every number, and it has a predictable average. This gives mathematicians a new way to understand how numbers behave when you start stacking them up in complex ways.

In short: They took a chaotic, weird mathematical function, applied a new high-tech lens, and discovered it's actually a very well-behaved, smooth, and predictable character.

1. Problem Statement and Motivation

The paper investigates the statistical properties of a specific arithmetic function, $S_s(n)$ , defined as the sum of the sum-of-proper-divisors function $s(d)$ over all divisors $d$ of $n$ :
$S_s(n) := \sum_{d|n} s(d)$
where $s(n) = \sigma(n) - n$ (the sum of proper divisors).

The study is motivated by the generalization of practical numbers. A number $n$ is $f$ -practical if every integer up to $S_f(n) = \sum_{d|n} f(d)$ can be represented as a sum of distinct values $f(d)$ . The authors focus on the case where $f(n) = s(n)$ .

The primary research questions are:

Does the normalized function $S_s(n)/n$ possess an asymptotic distribution function (a.d.f.)?
Is the range of values of $S_s(n)/n$ dense in the interval $[0, \infty)$ ?
What are the mean values and higher-order moments of $S_s(n)/n$ ?

2. Methodology and Tools

The authors employ tools from probabilistic number theory, specifically focusing on the distribution of arithmetic functions.

Decomposition: They utilize the identity $S_s(n) = S_\sigma(n) - \sigma(n)$ , where $\sigma(n)$ is the standard sum-of-divisors function and $S_\sigma(n) = \sum_{d|n} \sigma(d)$ . This allows them to relate $S_s(n)$ to known multiplicative functions.
Limitations of Classical Methods: The authors note that the classical Erdős-Wintner Theorem (which characterizes the existence of limiting distributions for additive functions) cannot be directly applied to $S_s(n)/n$ $S_{s} (n) / n$ because:
1. $S_s(n)/n$ is not multiplicative.
2. The logarithm of the related function, $\log(\sigma(n)/n)$ , has a purely singular distribution function, which complicates standard analytic approaches to proving continuity of the distribution.
Modern Approach (Lebowitz-Lockard and Pollack): To overcome the singularity issue, the paper highlights a novel method introduced by Lebowitz-Lockard and Pollack. This method relies on the concept of non-clustering:
- A function $f$ "clusters" around $x$ if the density of $n$ such that $f(n) \approx x$ is positive.
- If an arithmetic function has an a.d.f. and is non-clustering, its distribution function is continuous.
- The authors use theorems regarding linear combinations of multiplicative functions to prove non-clustering.
Moment Estimation: For mean value and moment calculations, the authors use Wintner's Mean Value Theorem for multiplicative functions and binomial expansion techniques to bound the growth of moments.

3. Key Contributions and Results

A. Density of Values (Section 3)

Theorem 3.1: The values of $S_s(n)/n$ are dense in the interval $[0, \infty)$ .

Method: Since $S_s(n)/n$ is not multiplicative, the authors cannot use Schoenberg's classical proof for $\phi(n)/n$ . Instead, they derive a recursive relationship for coprime integers $a, b$ :
$S_s(ab) = S_s(a)S_s(b) + \sigma(a)S_s(b) + \sigma(b)S_s(a)$
Using this, they construct a sequence of integers $N_k$ by iteratively multiplying by carefully chosen primes $q$ to steer the value of $S_s(N_k)/N_k$ arbitrarily close to any target $x \in [0, \infty)$ .

B. Existence and Continuity of Distribution Function (Section 4)

Theorem 4.4: The function $S_s(n)/n$ possesses a continuous asymptotic distribution function.

Method:
1. Express $S_s(n)/n = f_1(n) - f_2(n)$ where $f_1 = S_\sigma(n)/n$ and $f_2 = \sigma(n)/n$ .
2. Both $f_1$ and $f_2$ are multiplicative and known to have distribution functions.
3. By applying the Lebowitz-Lockard and Pollack Theorems, they show that the linear combination $F = f_1 - f_2$ is non-clustering.
4. By Lemma 4.1, if a function has an a.d.f. and is non-clustering, that distribution function is continuous. This bypasses the difficulty posed by the singular nature of $\log(\sigma(n)/n)$ .

C. Mean Values and Moments (Section 5)

Theorem 5.1 & Corollary 5.2: The mean value of $S_s(n)$ and $S_s(n)/n$ are computed explicitly:
$M(S_s(n)/n) = \zeta(2)(\zeta(2) - 1)$
where $\zeta(2) = \pi^2/6$ .

Theorem 5.4: The $k$ -th moments $\mu_k$ of $S_s(n)/n$ exist, are finite, and satisfy the growth bound:
$\log \mu_k \ll k \log \log k$

Method: The authors expand $(S_s(n)/n)^k$ using the Binomial Theorem into a sum of terms involving powers of $\sigma(n)$ and $S_\sigma(n)$ . They verify the convergence conditions of Wintner's Mean Value Theorem for these terms. They then bound the moments by comparing them to the moments of $(n/\phi(n))^2$ , utilizing known estimates for the latter.

4. Significance

Novelty in Distribution Theory: The paper's most significant contribution is the successful application of the Lebowitz-Lockard and Pollack framework to a function that is neither additive nor multiplicative. This demonstrates a powerful alternative to the Erdős-Wintner theorem when dealing with functions where classical analytic methods fail due to singular distribution components.
Extension of Practical Number Theory: By analyzing $S_s(n)$ , the authors deepen the understanding of the statistical behavior of generalized practical numbers, specifically those generated by the sum-of-proper-divisors function.
Statistical Characterization: The paper provides a complete statistical profile of $S_s(n)/n$ , establishing that it behaves "typically" (has a continuous distribution), can take on any positive real value (density), and has well-behaved moments, despite the erratic nature of the underlying divisor functions.

In summary, Aidun and Thompson successfully characterize the statistical landscape of $S_s(n)/n$ , proving its values are dense in $[0, \infty)$ , its distribution is continuous, and its moments are well-controlled, utilizing modern probabilistic tools to overcome the limitations of classical analytic number theory.