A generalization on the average ratio of the smallest and largest prime divisor of $n$

Biao Wang

arxiv: 1907.09129 · v1 · pith:HOUJSKIZnew · submitted 2019-07-15 · 🧮 math.NT

A generalization on the average ratio of the smallest and largest prime divisor of n

Biao Wang This is my paper

Pith reviewed 2026-05-24 21:09 UTC · model grok-4.3

classification 🧮 math.NT

keywords smallest prime divisorlargest prime divisorratio averagekth powerasymptotic estimateErdős-van LintJia's methodnumber theory

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The pith

An asymptotic estimate is given for the average of the kth power of the ratio between the smallest and largest prime divisors of n.

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

The paper generalizes a 1982 result of Erdős and van Lint that estimated the average value of the ratio of the smallest prime divisor of n to its largest prime divisor. It applies C.H. Jia's method to obtain a corresponding estimate when that ratio is raised to any positive integer power k. A reader would care because the extension shows that the same technique handles the powered case and therefore supplies information about higher moments of the ratio across integers up to x. The result concerns the sum over n ≤ x of (p_min(n)/p_max(n))^k and its asymptotic behavior.

Core claim

By applying C.H. Jia's method directly to the kth power, the paper derives an estimate for the average of the positive integer power of the ratio of the smallest and largest prime divisor of n, thereby generalizing the earlier Erdős-van Lint estimate.

What carries the argument

Application of C.H. Jia's method to the kth power of the smallest-to-largest prime divisor ratio.

If this is right

The same asymptotic form holds for every positive integer k.
The average order of the powered ratio can be estimated without new major analytic tools.
The result supplies information on higher moments of the prime-factor ratio across all n up to x.
The method of Jia remains applicable once the ratio is replaced by its kth power.

Where Pith is reading between the lines

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

Similar estimates might be obtainable for non-integer powers or for other symmetric functions of the two prime divisors.
The constants appearing in the estimate could be checked numerically for small fixed k by direct summation up to moderate x.
The extension suggests that other averages involving the prime divisors of n may admit powered versions under the same technique.

Load-bearing premise

C.H. Jia's method extends directly to produce a valid estimate when the ratio is raised to a positive integer power k.

What would settle it

A direct numerical computation of the sum up to a large x of (smallest prime factor of n over largest prime factor of n) raised to k, compared against the claimed asymptotic main term, would falsify the estimate if the two disagree by more than the error term.

read the original abstract

In 1982, Erd\"os and van Lint showed an estimate for the average of the ratio of the smallest and largest prime divisor of $n$. In this note, we apply C.H. Jia's method to give an estimate for the average of positive integer power of the ratio.

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.

Desk Editor's Note private letter to a colleague

Short note applies Jia's method to the k-th power average of p_min/p_max but provides no visible check that error terms remain uniform for k>1.

read the letter

The main takeaway is that this is a short note claiming an extension of the 1982 Erdős-van Lint average for the ratio of smallest to largest prime divisor to its positive integer powers, done by carrying over Jia's method. That is the entire contribution on offer. The abstract states the claim directly and cites the right prior results, which is fine as far as it goes. The work is honest in positioning itself as an application rather than a new technique. Beyond that there is not much to credit. The paper does not appear to introduce new ideas, new estimates with independent interest, or any verification that the method survives the change in weighting when the ratio is raised to k. The stress-test concern lands here: replacing the integrand with its k-th power shifts emphasis toward smaller ratios, and it is not automatic that the truncation or main-term extraction from the k=1 case absorbs the difference without fresh bounds. Since only the abstract is visible, we cannot see whether the paper re-derives the relevant sums or integrals or simply asserts the extension works. That leaves the central claim uncheckable at present. This kind of technical note is aimed at analytic number theorists who track averages over prime divisors and might want the powered version on record. A reader already familiar with Jia's work and the Erdős-van Lint result could extract the statement quickly, but the paper does not supply enough detail to be useful on its own. It is not incoherent or circular, so it shows basic engagement with the literature. I would send it to a referee rather than desk-reject if the full manuscript exists and the proof is written out, because the claim is narrow enough that a specialist can verify the extension in a few pages. Expect the referee to ask for explicit confirmation on the error terms for k greater than 1.

Referee Report

1 major / 0 minor

Summary. The manuscript extends the 1982 Erdős–van Lint estimate for the average value of p_-(n)/p_+(n) (smallest over largest prime divisor of n) to the average of [p_-(n)/p_+(n)]^k for each fixed positive integer k, by applying the method introduced by C.H. Jia.

Significance. If the extension is valid, the result supplies a modest generalization of an existing average-order statement in analytic number theory. Because the paper presents itself as a short note that applies an already-published method rather than deriving new estimates or error terms, its significance is limited to confirming that the same technique carries over to the powered ratio.

major comments (1)

[Abstract] The central claim—that Jia’s method applies directly and yields a valid estimate for the k-th power—requires explicit verification that the error terms remain uniform when the integrand or weighting is replaced by its k-th power. The abstract states the application but supplies no derivation, no modified integrals, and no uniformity statement in k; this is load-bearing for the result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] The central claim—that Jia’s method applies directly and yields a valid estimate for the k-th power—requires explicit verification that the error terms remain uniform when the integrand or weighting is replaced by its k-th power. The abstract states the application but supplies no derivation, no modified integrals, and no uniformity statement in k; this is load-bearing for the result.

Authors: The referee is correct that the abstract (and the short note format) contains no explicit derivation or modified integrals. For each fixed positive integer k the application of Jia’s method proceeds by replacing the integrand with its k-th power; the resulting error terms retain the same order as in the k=1 case, with implied constants now depending on k. No uniformity in k is asserted or required. To meet the referee’s concern we will add one clarifying sentence in the introduction stating that the estimates of Jia carry over directly for fixed k. revision: yes

Circularity Check

0 steps flagged

No circularity: applies external method of Jia to k-power case with independent citations

full rationale

The paper's central claim is an application of C.H. Jia's existing method (cited as prior independent work) to estimate the average of (p_min(n)/p_max(n))^k for positive integer k, building on the k=1 case of Erdős-van Lint 1982. The abstract and structure indicate a direct extension without re-deriving or fitting the core estimates from the paper's own data or definitions. No self-citations are load-bearing, no fitted inputs are relabeled as predictions, and no ansatz or uniqueness result is smuggled in via the author's prior work. The derivation chain remains self-contained against the referenced external results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, ad-hoc axioms, or invented entities are stated. The work relies on standard results from analytic number theory referenced via citations.

axioms (1)

standard math Standard results and methods from analytic number theory on prime divisors and averages, as used in Erdős-van Lint and Jia.
The note builds directly on these prior works without introducing new background assumptions in the abstract.

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