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

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.