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Claim. Let $Q(X)$ count the highly composite numbers in $[1,X]$. Nicolas, Théorème 4, proves

$$Q(X)=O\bigl((\log X)^{1+c}\bigr)$$

for an absolute constant $c>0$, the constant of his Théorème 3. So $Q(x)\gg_k(\log x)^k$ fails for every $k>1+c$, and the answer to Problem 381 is no. Théorème 3 counts the highly composite numbers lying between two consecutive superior highly composite numbers $N<N'$ and finds $O((\log N)^c)$ of them; the count up to $X$ follows by summing over the superior highly composite numbers below $X$. The local count rests on Théorème 1, a bound on how far $d(A)/A^{\varepsilon}$ can fall below $d(N)/N^{\varepsilon}$ for a highly composite number $A$, where $N$ is the superior highly composite number below $A$ at which that ratio is largest (Nicolas's bénéfice of $A$, at most $Cx^{-\gamma}$ with $x=2^{1/\varepsilon}$), proved through Diophantine approximation of the numbers $\log(1+1/k)/\log 2$ with Feldman's refinement of Baker's theorem on linear forms in logarithms. Théorème 5 of the same paper reproves Erdős's 1944 lower bound $Q(X)\ge(\log X)^{1+c'}$ with a larger constant, so the count is pinned between two fixed powers of $\log X$; Nicolas conjectures the exact exponent, $\log Q(X)/\log\log X\to1+(\log(3/2)+\log(5/4))/(4\log 2)=\log30/\log16 =1.2267\ldots$ (the print gives the value as 1.277 on p. 117, a misprint of the exact expression), which no result recorded here settles. The source card is Nicolas 1971; the lower bound it reproves is the accepted partial claim Erdős 1944 (card), where Erdős records that he could not decide the question this page answers.

Acceptance. Refereed: Canadian J. Math. 23 (1971), no. 1, 116–130 (February 1971 issue; the publication record gives no day, and the page's date is the first day of that month). Reviewed: the site's curator, T. F. Bloom, marks Problem 381 disproved and credits this paper for the answer. No formalization is recorded, and this repository has not checked the proof independently.