Unbounded logarithmic limsup in Erd\H{o}s problem 684

TL;DR

This paper proves that the function f(n) related to Erdős problem 684 exceeds any fixed multiple of log n infinitely often, refuting the expected upper bound and establishing that f(n) grows faster than log n.

Contribution

The authors resolve Erdős problem 684 at the order level, demonstrating that f(n) surpasses any constant times log n infinitely often using a novel multiplier sieve construction.

Findings

f(n) exceeds (C-o(1)) log n for any fixed C>1 infinitely often

limsup of f(n)/log n is infinite, disproving the conjectured upper bound

f(n) is shown to grow strictly faster than log n infinitely often

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.