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Statement

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

Théorème 5 (p. 127). There is a constant $c'>0$ such that $Q(X)\ge(\log X)^{1+c'}$.

The paper says (p. 127) that Erdős proved the theorem (reference [2]) and that it obtains a slightly larger $c'$ by essentially the same method. The proof (pp. 128–129) shows that every highly composite $A$ is followed by a highly composite $A''$ with

$$A<A''\le A\Bigl(1+\frac1{(\log A)^{c'}}\Bigr)$$

for any $c'<\tfrac13(\theta+\theta')(1-\tau)=0.113\ldots$, where $\theta=\log(3/2)/\log2$, $\theta'=\log(5/4)/\log2$ and $\tau=5/8$; Erdős had this gap bound with $c'=(1-\tau)/4=3/32$ (p. 129). Display (5) of the introduction (p. 117) records Erdős's bound as $Q(X)\ge(\log X)^{1+c}$ with $c=\tfrac14(1-\tau)\le3/32$.

Proof pointer

Pp. 127–129. Dirichlet's pigeonhole principle applied to the fractional parts $\{u\theta+v\theta'\}$, $\lvert u\rvert\le U$, $\lvert v\rvert\le V$, gives integers $u,v,w$ with $0<u\theta+v\theta'+w\le1/(4UV)$ (display (22)). From $A$ the paper builds $A'$ with $\log d(A')=\log d(A)+(u\theta+v\theta'+w)\log2$ by moving primes across the largest primes of $A$ with exponents $4$, $2$ and $1$ (located by Proposition 4 near $x_4=x^{\theta'}$, $x_2=x^\theta$ and $x$). Displays (13), (15) and (12) bound $\epsilon\log(A'/A)$, and the choice $U=x^\alpha$, $V=x^\beta$ with $\alpha=\tfrac13(2\theta-\theta')(1-\tau)$ and $\beta=\tfrac13(2\theta'-\theta)(1-\tau)$ gives the gap bound; since $d(A')>d(A)$, a highly composite number lies in $(A,A']$.

Dependencies

The paper's Proposition 4 (p. 120) and displays (12), (13), (15) (pp. 118–120); Ingham's theorem on primes in short intervals (display (4), p. 116). Erdős's earlier proof: Erdős 1944, Theorem.

Read depth

Claims checked: the statement, the constants on pp. 128–129 and display (5) were read on the page images of the print. The proof was read for its structure and is not reconstructed or independently reviewed here.

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 theorem gives $Q(x)\ge(\log x)^{1+c'}$, so the bound the problem asks for holds for the exponents $k\le1+c'$; it does not decide the question, which Théorème 4 answers.