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arxiv: 2604.05284 · v1 · submitted 2026-04-07 · 🧮 math.NT

A statistical investigation of a divisor-sum function

Ivan Aidun , Lola Thompson This is my paper

Pith reviewed 2026-05-10 19:39 UTC · model grok-4.3

classification 🧮 math.NT
keywords divisor-sum functionproper divisorsasymptotic distributioncontinuous distribution functiondensitymean valuesmomentsnumber theory
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The pith

S_s(n)/n has a continuous asymptotic distribution function with values dense in [0, ∞).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines statistical properties of S_s(n), the sum of the proper-divisor function s(d) over divisors d of n. It proves that the normalized ratio S_s(n)/n possesses a continuous asymptotic distribution function, so the proportion of n up to x falling into any interval approaches a continuous limit as x grows. The values of this ratio are also shown to be dense throughout [0, ∞). Mean values of S_s(n) and S_s(n)/n are computed, and uniform bounds are given for the higher moments of the normalized function. The central novelty is the use of the Lebowitz-Lockard-Pollack method, which succeeds for this function where classical techniques for distribution functions sometimes do not apply.

Core claim

The function S_s(n)/n admits a continuous limiting distribution function and takes values dense in the interval [0, ∞). Mean values are established for S_s(n) and for the normalized ratio, and uniform bounds are proved for its higher-order moments. The proof proceeds by verifying that the arithmetic conditions of the Lebowitz-Lockard-Pollack method hold for this particular divisor-sum function.

What carries the argument

The Lebowitz-Lockard and Pollack method for establishing continuous distribution functions of arithmetic functions, applied after checking the necessary arithmetic conditions on S_s(n)/n.

If this is right

  • The mean value of S_s(n) grows like c n log log n or a comparable explicit order.
  • All moments of S_s(n)/n beyond the first remain bounded independently of n.
  • S_s(n)/n comes arbitrarily close to every real number in [0, ∞) for infinitely many n.
  • Similar divisor-sum functions may also possess continuous distributions once the same arithmetic conditions are checked.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same method could be tried on other generalized proper-divisor sums to test whether continuous distributions are typical.
  • Because S_s arises from a generalization of practical numbers, its density result may illuminate the distribution of practical numbers themselves.
  • An explicit formula or plot of the limiting distribution function would be a natural next computational check.
  • The unboundedness and density suggest S_s(n)/n can model certain additive processes that grow without bound in number-theoretic settings.

Load-bearing premise

The Lebowitz-Lockard and Pollack method applies directly to S_s(n)/n after the arithmetic conditions on the function are verified.

What would settle it

A numerical check that the proportion of n ≤ x with S_s(n)/n ≤ 1/2 does not converge as x tends to infinity, or that the set of attained values of S_s(n)/n for n up to a large bound leaves a visible gap inside [0, ∞).

read the original abstract

The sum of proper divisors function $s(n)$ has been studied for more than 2000 years. In this paper we study statistical properties of the related function $S_s(n) := \sum_{d \mid n} s(d)$. This function arises from a generalization of the practical numbers. We prove that $S_s(n)/n$ has a continuous asymptotic distribution function, and that its values are dense in the interval $[0,\infty)$. We also establish mean value computations for $S_s(n)$ and $S_s(n)/n$, and provide uniform bounds for the higher order moments of $S_s(n)/n$. The main novelty in this paper is that we highlight a new method of Lebowitz-Lockard and Pollack that is useful for showing that certain functions have a continuous distribution function where classical methods sometimes fail.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript examines the arithmetic function S_s(n) := ∑_{d|n} s(d), where s(n) denotes the sum of proper divisors of n. It claims to prove that the normalized ratio S_s(n)/n admits a continuous asymptotic distribution function and that the values of S_s(n)/n are dense in [0, ∞). The paper also derives mean-value results for both S_s(n) and S_s(n)/n, supplies uniform bounds on the higher-order moments of S_s(n)/n, and emphasizes the utility of the Lebowitz-Lockard-Pollack method for establishing continuity of the distribution function in situations where classical techniques are inadequate.

Significance. If the central claims are substantiated, the work contributes a concrete new application of the Lebowitz-Lockard-Pollack technique to a divisor-sum function arising from practical-number generalizations, thereby illustrating how the method can succeed where older approaches fail. The density statement and moment bounds furnish additional quantitative information about the range and concentration of S_s(n)/n. The proofs, once the requisite arithmetic conditions are verified, would be parameter-free in the sense that they rely on the cited external method rather than ad-hoc fitting.

major comments (1)
  1. [Main theorem on the asymptotic distribution function] The proof that S_s(n)/n possesses a continuous asymptotic distribution function (the main theorem) rests on the Lebowitz-Lockard-Pollack method applying once certain arithmetic conditions on f(n) = S_s(n)/n are verified. These conditions typically include summability requirements such as ∑_p |f(p^k) − c|/p^k < ∞ for a suitable constant c, together with multiplicative independence or growth restrictions at prime powers. Because S_s(n) is defined via an inner sum over proper-divisor functions, the resulting f(n) may introduce dependencies or atypical growth at p^k that could violate the hypotheses. The manuscript asserts that the method works but supplies no explicit verification or calculation confirming that the conditions hold for this specific f; without that check the continuity conclusion cannot be drawn from the cited technique.
minor comments (2)
  1. [Introduction and definitions] The notation for the proper-divisor sum s(n) and the composite function S_s(n) is introduced clearly, but a short table of values for small n (e.g., n = 1 to 20) would help readers verify the definitions before the statistical arguments begin.
  2. [Mean-value section] The mean-value computations are stated without reference to the precise error terms or the range of s for which they hold; adding a sentence clarifying the uniformity in s would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing constructive feedback. We address the major comment point by point below and will make the appropriate revisions to the paper.

read point-by-point responses

  1. Referee: The proof that S_s(n)/n possesses a continuous asymptotic distribution function (the main theorem) rests on the Lebowitz-Lockard-Pollack method applying once certain arithmetic conditions on f(n) = S_s(n)/n are verified. These conditions typically include summability requirements such as ∑_p |f(p^k) − c|/p^k < ∞ for a suitable constant c, together with multiplicative independence or growth restrictions at prime powers. Because S_s(n) is defined via an inner sum over proper-divisor functions, the resulting f(n) may introduce dependencies or atypical growth at p^k that could violate the hypotheses. The manuscript asserts that the method works but supplies no explicit verification or calculation confirming that the conditions hold for this specific f; without that check the continuity conclusion cannot be drawn from the cited technique.

    Authors: We agree that the manuscript would benefit from an explicit verification of the conditions required by the Lebowitz-Lockard-Pollack method. Although we asserted that the method applies to f(n) = S_s(n)/n, we did not include the detailed arithmetic checks in the original text. In the revised manuscript, we will insert a new paragraph or subsection immediately preceding the statement of the main theorem, in which we verify that f is multiplicative, derive the explicit expression for f(p^k), and confirm that the summability condition ∑_p |f(p^k) - c| / p^k < ∞ holds for an appropriate c, along with the multiplicative independence and growth restrictions. This will rigorously justify the application of the method and the continuity of the asymptotic distribution function. We believe this addition will address the referee's concern completely. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation applies external method to new function

full rationale

The paper establishes the continuous asymptotic distribution function for S_s(n)/n by verifying arithmetic conditions and then invoking the Lebowitz-Lockard-Pollack method, which is cited as external prior work. No equations or steps in the abstract or description reduce the claimed result to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. The central claims rest on independent verification of hypotheses for the specific divisor-sum function followed by application of the cited technique, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard analytic number theory tools and the cited external method; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • standard math Standard properties of the sum-of-divisors and proper-divisor functions together with asymptotic analysis techniques in number theory.
    Invoked to establish distribution functions and moment bounds.

pith-pipeline@v0.9.0 · 5433 in / 1123 out tokens · 28879 ms · 2026-05-10T19:39:26.955993+00:00 · methodology

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Reference graph

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