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Statement

Question (p. 130, unnumbered, quoted). "At present I cannot decide whether the number of highly composite numbers not exceeding $x$ is greater than $(\log x)^k$ for every $k$."

Here $n$ is highly composite if $d(m)<d(n)$ for all $m<n$, with $d$ the divisor function. The sentence follows the paper's announcement of the lower bound $(\log x)^{1+c}$ for a certain $c$, proved through the Theorem of p. 131; the paper does not answer the question.

Source. P. Erdős, On highly composite numbers, J. London Math. Soc. 19 (1944), 130–133, p. 130. The edition read is named on the source card.

Read depth. Claims checked: the sentence was read on the page image of the print. Nothing here is independently reviewed.

Proof pointer

None: the paper poses the question without a proof or a sketch.

Dependencies

None.

Bears on

• Problem 381: the problem asks whether $Q(x)\gg_k(\log x)^k$ for every $k\ge1$, with $Q(x)$ the number of highly composite numbers in $[1,x]$. That is the question posed here, in the form "greater than $(\log x)^k$ for every $k$"; this page records the 1944 posing only, and the answers are on the problem page.