On the digits of the sum of proper divisors

Technical Summary: On the Digits of the Sum of Proper Divisors

Problem Statement
This paper investigates the probabilistic behavior of the decimal (and general base-$g$) digits of $s(n)$, the sum of the proper divisors of a positive integer $n$. The authors address three primary questions:

1. Digit Occurrence: How frequently do all possible digits appear in the leading and trailing positions of $s(n)$?
2. Leading Digit Distribution: Do the leading digits of $s(n)$ follow Benford's law?
3. Sparse Preimages: How many composite integers $n$ exist such that $s(n)$ is "ellipsephic" (i.e., missing at least one digit in its base-$g$ expansion)?

The study is motivated by the contrast between the behavior of $s(n)$ on all integers versus its behavior when restricted to composite integers, specifically regarding the preimages of sets with missing digits.

Methodology
The authors employ a combination of analytic number theory, probabilistic methods, and sieve techniques.

• Digit Occurrence (Theorems 1.1, 2.2, 2.5): To prove that almost all integers $n \le x$ have $s(n)$ containing all digits in the first and last $k(x)$ positions (where $k(x) \to \infty$), the authors utilize:

  • Modular Constraints: They analyze $s(n) \pmod{g^k}$ and relate this to $\sigma(n) \pmod{g^k}$.
  • Smooth Numbers: They decompose $n$ into a smooth part and a rough part, utilizing the Prime Number Theorem to bound the contribution of exceptional cases.
  • Distribution of $\sigma(n)/n$: They rely on the continuity of the distribution function of $\sigma(n)/n$ (Davenport's results) to show that $s(n)$ behaves like a random integer of comparable size regarding digit distribution.

• Benford's Law (Theorems 1.2, 3.5, 3.6):

  • Logarithmic Density: The authors prove $s(n)$ obeys Benford's law with respect to logarithmic density. They apply Weyl's criterion for uniform distribution modulo 1 to the sequence $\{\log_g s(n)\}$.
  • Halász's Theorem: Since $s(n)$ is not multiplicative, they express $s(n)^{i\alpha}$ using a binomial expansion of $\sigma(n)^{i\alpha}(1 - n/\sigma(n))^{i\alpha}$. They truncate this series and apply a weighted version of Halász's Theorem (Proposition 3.3) to show the logarithmic mean of the resulting multiplicative functions is zero.
  • Natural Density: Conversely, they prove $s(n)$ does not obey Benford's law with respect to natural density. They construct a specific set of "convenient" integers (multiples of 6 with large prime factors) where $s(n)$ is tightly clustered near $n$, creating a bias in the fractional parts of $\log_g s(n)$ that violates the uniform distribution required for natural density Benford behavior.

• Composite Integers and Missing Digits (Theorem 1.5):

  • Exclusion of Primes: The paper addresses a gap in previous work (Benli et al., 2026) which established an upper bound for $s(n)$ missing digits but noted that prime inputs ($s(p)=1$) dominate the count.
  • Decomposition: For composite $n$, they write $n = Pm$ where $P = P^+(n)$ is the largest prime factor. They analyze $s(n) = P s(m) + \sigma(m)$.
  • Sieve and Smoothness: They split the analysis based on the size of $m$ and $P$. For large $P$, they use the Brun-Titchmarsh theorem and bounds on the number of integers with missing digits (ellipsephic numbers) to show the count is significantly smaller than the general case.
  • Modular Arithmetic: They utilize Lemma 4.3 to bound the number of $m$ such that $s(m)$ is divisible by specific moduli, controlling the distribution of $s(n)$ modulo $g^k$.

Key Results

1. Universal Digit Occurrence: For any base $g \ge 2$ and any function $k(x) \to \infty$, asymptotically 100% of integers $n \le x$ have $s(n)$ containing all $g$ digits in both the first $k(x)$ and last $k(x)$ positions (Theorem 1.1).
2. Benford's Law (Logarithmic Density): The function $s(n)$ satisfies Benford's law with respect to logarithmic density. Specifically, the logarithmic density of $n$ for which the leading digits of $s(n)$ form a block $D$ is $\log_g(1 + 1/D)$ (Theorem 1.2).
3. Failure of Benford's Law (Natural Density): $s(n)$ does not satisfy Benford's law with respect to natural density (Proposition 3.6).
4. Composite Preimages of Sparse Sets: When restricting $n$ to composite numbers, the number of $n \le x$ such that $s(n)$ misses a specific digit $a_0$ in base $g$ is bounded by $O(x \exp(-c\sqrt{\log x}))$ for some constant $c > 0$ (Theorem 1.5). This is a significantly stronger bound than the $O(x \exp(-(\log \log x)^\gamma))$ bound for all integers, highlighting that prime inputs are the primary source of "missing digit" values for $s(n)$.

Significance and Claims
The paper claims to establish that the digits of $s(n)$ exhibit a high degree of randomness, behaving similarly to random integers in terms of digit occurrence and leading digit distribution (under logarithmic density).

A central contribution is the clarification of the role of prime numbers in the preimage of sets with missing digits. The authors demonstrate that while the set of all integers $n$ where $s(n)$ has missing digits is relatively large (due to the trivial case $s(p)=1$), the subset of composite integers with this property is much smaller. This result refines the understanding of the Erdős–Granville–Pomerance–Spiro (EGPS) conjecture, which posits that the preimage of a set with zero asymptotic density also has zero asymptotic density. The authors show that for the specific case of missing digits, the "density" of the preimage is driven almost entirely by primes, and excluding them yields a much sparser set.

The work relies on standard tools from analytic number theory (Halász's theorem, Brun-Titchmarsh, distribution of $\sigma(n)/n$) and does not propose new experimental applications or future implications beyond the theoretical bounds established.