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Statement

Setting. $Q(X)$ is the number of highly composite numbers less than $X$.

Conjecture (p. 117, unnumbered). The paper conjectures that

$$\lim_{X\to\infty}\frac{\log Q(X)}{\log\log X} =1+\frac{\log(3/2)+\log(5/4)}{4\log2}.$$

The print gives the value of the right side as $1.277\ldots$, a misprint. The expression equals $1+\tfrac14(\theta+\theta')$ with $\theta=\log(3/2)/\log2$ and $\theta'=\log(5/4)/\log2$, which is $\log30/\log16=1.2267\ldots$. The conclusion (p. 129) likewise prints $\tfrac14(\theta+\theta')=0.277\ldots$, where the value is $0.2267\ldots$, and the same $1.277\ldots$ on p. 130.

Basis given in the paper

Section 5 (pp. 129–130) gives the heuristic. If primes were spaced about $\log x$ apart near $x$, the argument of Théorème 5 would give $c'=\tfrac14(\theta+\theta')$. If the linear forms satisfied $\lvert u\theta+v\theta'+w\rvert>1/(\overline K(\eta)(uv)^{1+\eta})$ for every $\eta>0$, the paper says Théorème 1 would hold with $\gamma=\tfrac14(\theta+\theta')-\eta$ and the proof of Théorème 3 with $k'=5$; since $\log(4/3)/\log2=1-\theta$ and $\log(6/5)/\log2=\theta-\theta'$, only $k=2$ and $k=4$ would contribute, giving $(\log X)^{c-\eta}\le Q(X)\le(\log X)^{c+\eta}$ with $c=1+\tfrac14(\theta+\theta')$. The paper adds that if the numbers $\{u\theta+v\theta'\}$ were badly distributed, $\log Q(X)/\log\log X$ would probably have no limit. None of this is proved in the paper.

Read depth

Claims checked: the conjecture and Section 5 were read on the page images of the print, and the numerical values were recomputed from the printed expressions. Not independently reviewed.

Source. Jean-Louis Nicolas, Répartition des nombres hautement composés de Ramanujan, Canadian J. Math. 23 (1971), no. 1, 116–130, doi:10.4153/cjm-1971-012-6; the edition read is named on the source card.

Bears on

• Problem 381: the conjecture predicts the exponent of growth of $Q(X)$ that the problem's question concerns. It is a conjecture, not a result, and the problem's answer rests on Théorème 4.