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arxiv: 2604.23784 · v2 · submitted 2026-04-26 · 🧮 math.NT · math.CO

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Unbounded logarithmic limsup in ErdH{o}s problem 684

Ji Ho Bae
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Pith reviewed 2026-05-08 05:19 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords Erdős problem 684binomial coefficient factorizationlimsup f(n)/log nFourier sievearithmetic progressionsBurgess estimatelcm construction
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The pith

The function f(n) from Erdős problem 684 satisfies limsup f(n)/log n = infinity.

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

The paper shows that f(n), defined as the smallest k where the small-prime part of binomial(n,k) exceeds n squared, can exceed any fixed multiple of log n for infinitely many n. It constructs such n as t times the least common multiple of 1 to M, minus one, where t is a small multiplier obtained from a Fourier sieve. This construction reduces the problem to an arithmetic-progression estimate that combines Timofeev's mean-value results with Burgess-type savings. The result refutes the long-expected bound that f(n) remains much smaller than log n, while a separate upper bound of order (log n) squared already exists.

Core claim

By a short-multiplier construction n_M = t L_M -1, where L_M = lcm(1,...,M) and t is a multiplier of size exp(o(M)) extracted from a Fourier sieve, we prove that for every fixed C>1 there exist integers n with f(n)>(C-o(1)) log n, hence limsup f(n)/log n = infinity.

What carries the argument

short-multiplier construction n_M = t L_M -1 with t of size exp(o(M)) from a Fourier sieve, reduced to a dyadic fixed-Omega arithmetic-progression estimate via Q_M = M!/L_M box parametrization, local harmonic-height cap, and exact-a product-shell extraction, using Timofeev's framework plus Burgess mod-p saving

If this is right

  • For any fixed C>1 the inequality f(n) > (C-o(1)) log n holds for infinitely many n.
  • The conjectured bound f(n) << log n fails for all implied constants.
  • The growth of f(n) lies strictly above log n infinitely often and at most (log n)^2 for all n.

Where Pith is reading between the lines

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

  • The same multiplier-sieve technique could be tested on related Erdős questions that track prime factors in binomial coefficients.
  • If the upper bound of (log n)^2 is not sharp, the true order of f(n) may involve an extra iterated logarithm factor.
  • The reduction step isolates a precise arithmetic-progression mean-value problem that might admit further improvements via stronger exponential-sum estimates.

Load-bearing premise

A Fourier sieve extracts a multiplier t of size exp(o(M)) for which the dyadic fixed-Omega arithmetic-progression estimate produces the claimed lower bound on f(n).

What would settle it

An explicit large-M computation showing that every candidate multiplier t of size exp(o(M)) produces an n_M where f(n_M) remains bounded by some fixed multiple of log n_M would disprove the limsup claim.

read the original abstract

For $0\le k\le n$, write $\binom nk=uv$ where the primes dividing $u$ are at most $k$ and the primes dividing $v$ exceed $k$, and let $f(n)$ be the least $k$ with $u>n^2$; Erd\H{o}s problem 684 asks for bounds on $f(n)$. We resolve the problem at the order level. By a short-multiplier construction $n_M=tL_M-1$, where $L_M=\operatorname{lcm}(1,\ldots,M)$ and $t$ is a multiplier of size $\exp(o(M))$ extracted from a Fourier sieve, we prove that for every fixed $C>1$ there exist integers $n$ with $$ f(n)>(C-o(1))\log n, $$ hence $$ \limsup_{n\to\infty}\frac{f(n)}{\log n}=\infty. $$ We thus refute the widely expected upper bound $f(n)\ll\log n$ and place the order of $f(n)$ strictly above $\log n$ infinitely often. A matching polylogarithmic upper bound $f(n)\ll(\log n)^2$ is known by Alexeev, Putterman, Sawhney, Sellke, and Valiant (arXiv:2603.29961). The reduction of the multiplier sieve to a dyadic fixed-$\Omega$ arithmetic-progression estimate, including a $Q_M=M!/L_M$ box parametrization, a local harmonic-height cap, and an exact-$a$ product-shell extraction, is new. The required estimate uses Timofeev's mean-in-progressions framework together with a Burgess-based mod-$p$ saving on the relevant prime band.

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

2 major / 3 minor

Summary. The manuscript resolves Erdős problem 684 at the order level. It defines f(n) as the smallest k for which, in the factorization binom(n,k)=uv with all prime factors of u at most k and those of v exceeding k, one has u>n². Using the short-multiplier construction n_M = t L_M -1 with L_M=lcm(1..M) and t=exp(o(M)) extracted via a Fourier sieve, together with a new reduction to a dyadic fixed-Ω arithmetic-progression estimate (via the Q_M=M!/L_M box parametrization, local harmonic-height cap, and exact-a product-shell extraction), the authors apply Timofeev's mean-in-progressions framework and Burgess mod-p saving to prove that f(n)>(C-o(1))log n for every fixed C>1, hence limsup f(n)/log n=∞. A matching upper bound f(n)≪(log n)² is cited from prior work.

Significance. If the central claim holds, the result is significant: it refutes the long-expected bound f(n)≪log n and shows that the order of f(n) lies strictly above log n for infinitely many n, while remaining below the known polylogarithmic upper bound. The new reduction techniques (Fourier-sieve multiplier extraction, Q_M-box parametrization, and product-shell extraction) constitute a concrete methodological advance that may be reusable in other problems involving the prime factors of binomial coefficients.

major comments (2)
  1. [Reduction to arithmetic-progression estimate] The reduction step from the Fourier-sieve multiplier t to the dyadic fixed-Ω arithmetic-progression estimate (via Q_M=M!/L_M, local harmonic-height cap, and exact-a product-shell extraction) is load-bearing for the lower bound; the manuscript must supply explicit error-term estimates showing that the o(M) size of t is preserved under these operations and that the resulting AP mean-value problem lies inside the range where Timofeev's framework plus Burgess saving applies without additional losses.
  2. [Application of Burgess saving] The applicability of Burgess bounds on the precise prime band arising after the product-shell extraction requires a quantitative statement of the saving obtained (including the admissible range of the modulus and the implied constant); without this, it is not immediate that the final lower bound reaches (C-o(1))log n for arbitrary fixed C.
minor comments (3)
  1. The abstract and introduction should include a brief numerical example computing f(n) for a small n (e.g., n=10 or n=20) to illustrate the definition.
  2. [Introduction] Ensure the reference to the matching upper bound (arXiv:2603.29961) is cited explicitly in the introduction and that the citation list is complete.
  3. [Fourier sieve construction] Notation for the Fourier sieve (in particular the precise meaning of 'short multiplier' and the harmonic-height cap) should be introduced with a short self-contained paragraph before the main construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for the positive evaluation of the significance of our result. We address the two major comments below and agree that both require additional explicit quantitative details in the manuscript. We will incorporate these in the revised version.

read point-by-point responses

  1. Referee: [Reduction to arithmetic-progression estimate] The reduction step from the Fourier-sieve multiplier t to the dyadic fixed-Ω arithmetic-progression estimate (via Q_M=M!/L_M, local harmonic-height cap, and exact-a product-shell extraction) is load-bearing for the lower bound; the manuscript must supply explicit error-term estimates showing that the o(M) size of t is preserved under these operations and that the resulting AP mean-value problem lies inside the range where Timofeev's framework plus Burgess saving applies without additional losses.

    Authors: We agree that the reduction is load-bearing and that the manuscript currently presents the outline of the steps without fully explicit error-term control. In the revision we will add a dedicated subsection providing the required estimates: we will show that the Q_M-box parametrization, local harmonic-height cap, and exact-a product-shell extraction each contribute an o(M) term to the exponent, so that the extracted multiplier remains of size exp(o(M)). We will further verify that the resulting dyadic fixed-Ω arithmetic-progression mean-value problem lies strictly inside the range of applicability of Timofeev's framework together with Burgess saving, with no losses that would obstruct the (C-o(1))log n lower bound for arbitrary fixed C>1. revision: yes

  2. Referee: [Application of Burgess saving] The applicability of Burgess bounds on the precise prime band arising after the product-shell extraction requires a quantitative statement of the saving obtained (including the admissible range of the modulus and the implied constant); without this, it is not immediate that the final lower bound reaches (C-o(1))log n for arbitrary fixed C.

    Authors: We agree that an explicit quantitative statement of the Burgess saving is needed. The manuscript applies Burgess bounds to the prime band after product-shell extraction, but the admissible modulus range and implied constant are left implicit. In the revision we will insert a self-contained lemma stating the precise Burgess estimate used, including the range of the modulus (matching the dyadic intervals and product-shell size) and the implied constant. This will confirm that the saving is sufficient to absorb all preceding error terms and to reach f(n)>(C-o(1))log n for every fixed C>1. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit construction plus external theorems

full rationale

The derivation proceeds by an explicit short-multiplier construction n_M = t L_M -1 (L_M = lcm[1..M]) where t = exp(o(M)) is obtained from a Fourier sieve, followed by a reduction (via the new Q_M = M!/L_M box parametrization, local harmonic-height cap, and exact-a product-shell extraction) to a dyadic fixed-Omega arithmetic-progression estimate. This estimate is then supplied by Timofeev's mean-in-progressions framework together with Burgess mod-p saving. No step equates the target lower bound f(n) > (C-o(1)) log n to a fitted parameter, a self-referential definition, or a load-bearing self-citation; the o(M) size is compatible with log n ~ M and the implication follows directly once the AP estimate is granted. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the Fourier-sieve multiplier extraction and the applicability of Timofeev's mean-in-progressions estimates with Burgess saving to the specific dyadic fixed-Omega setting created by the Q_M parametrization. No free parameters are fitted to data; the o(M) size of t is derived from the sieve. No new entities are postulated.

axioms (2)
  • domain assumption Timofeev's mean-in-progressions framework together with Burgess-based mod-p saving applies to the arithmetic progressions arising after the Q_M box parametrization and local harmonic-height cap.
    Invoked to obtain the required estimate on the multiplier sieve.
  • domain assumption The Fourier sieve produces a multiplier t of size exp(o(M)) compatible with the n_M = t L_M -1 construction and the exact-a product-shell extraction.
    Central to achieving the (C-o(1)) log n lower bound for arbitrary C.

pith-pipeline@v0.9.0 · 5620 in / 1763 out tokens · 102240 ms · 2026-05-08T05:19:41.700534+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

8 extracted references · 2 canonical work pages · 1 internal anchor

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