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Claim. The Theorem of P. Erdős, On highly composite numbers, J. London Math. Soc. 19 (1944), 130–133 (p. 131): there is a positive constant $c$ such that, if $n$ is highly composite, then some highly composite number $n_1$ satisfies $n<n_1<n+n(\log n)^{-c}$. The proof (p. 132) rests on Ingham's theorem that for large $x$ the interval $(x,x+x^{5/8})$ contains about $x^{5/8}/\log x$ primes, and allows any exponent $c<3/32$. It follows at once (p. 130) that the number of highly composite numbers not exceeding $x$ is greater than $(\log x)^{1+c}$, which improves Ramanujan's bound $c\log x(\log\log x)^2(\log\log\log x)^{-2}$. The source card is Erdős 1944.

Covers. The exponents $1\le k\le1+c$ of Problem 381, with $k$ real as the Nicolas page reads it, answered yes: $Q(x)\gg(\log x)^k$ holds for every such $k$. In the same paper Erdős writes that he cannot decide whether $Q(x)$ exceeds $(\log x)^k$ for every $k$, which is the question, and Nicolas 1971 answers it no with the upper bound $Q(x)\ll(\log x)^{1+c'}$.

Depends on. Nothing in this wiki; the result rests on the cited paper alone.

Acceptance. Refereed: J. London Math. Soc. 19 (1944), 130–133, received 14 February 1944 and read 15 June 1944, in the issue of July 1944; the publication record gives no day, so the first day of that month stands in. The site's DISPROVED label credits Nicolas's upper bound, and its commentary records this bound as Erdős's without settling the problem by it, so no reviewed evidence is listed.