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Statement

Setting (p. 130). With $d(n)$ the number of divisors of $n$, a number $n$ is highly composite (Ramanujan's definition) if $d(m)<d(n)$ for all $m<n$. Throughout the paper $c$ denotes a positive absolute constant, not always the same one (footnote, p. 130).

Theorem (p. 131, quoted). "There is a positive constant $c$ such that, if $n$ is highly composite, then there is a highly composite number $n_1$ satisfying

$$n<n_1<n+n(\log n)^{-c}.$$

"

Counting consequence (p. 130). The number of highly composite numbers not exceeding $x$ is greater than $(\log x)^{1+c}$ "for a certain $c$". The paper says this "follows immediately" from the Theorem and gives no further argument. It improves Ramanujan's lower bound, recalled on p. 130, that the count exceeds $c\log x\,(\log\log x)^{1/2}(\log\log\log x)^{-3/2}$.

Proof pointer

Pp. 131–132. Write $n=2^{\kappa_2}3^{\kappa_3}\cdots p^{\kappa_p}$, so $p$ is the largest prime factor of $n$, and let $q$ be the largest prime with $\kappa_q\ge2$; Lemmas 2 and 3 place $q$ in $(4\sqrt p,\tfrac12p)$. Writing $q=p^\delta$, Dirichlet's approximation theorem gives positive integers $s,t$ with $s<p^{3/32}$ and $\lvert s\delta-t\rvert<p^{-3/32}$ (display (1)). Two candidates are compared: $n_1$ divides out one factor of each of the $s$ primes just below $q$ and multiplies in the $t$ primes just above $p$; $n_2$ divides out the $t$ primes just below $p$ and multiplies in the $s$ primes just above $q$. Display (2) and the lemmas fix the exponents involved, and the divisor counts show that one of the two has at least $d(n)$ divisors, hence exceeds $n$. Ingham's theorem keeps all the primes used within $q^{5/8}$ of $q$ or $p^{5/8}$ of $p$, which bounds the candidate by $n(1+p^{-\alpha})$ for any absolute constant $\alpha<3/32$ (p. 132); Lemma 1 ($p>c\log n$) turns this into display (3). The lemmas are proved on pp. 132–133 by similar exchanges of prime factors, using Bertrand's postulate and the prime number theorem.

Dependencies

None in the corpus. Inputs named by the paper:

• Ingham's improvement on Hoheisel's theorem (Quart. J. Math. Oxford 8 (1937), 255–266), stated on p. 130: for sufficiently large $x$ the number of primes in $(x,x+x^{5/8})$ is asymptotic to $cx^{5/8}(\log x)^{-1}$. The footnote on p. 130 adds that Hoheisel's original theorem, with an unspecified constant less than $1$ in place of $5/8$, would suffice for the main result.
• Dirichlet's approximation theorem (cited from Hardy and Wright).
• The paper's three lemmas (p. 131), stated for a sufficiently large highly composite $n=2^{\kappa_2}3^{\kappa_3}\cdots p^{\kappa_p}$, for which $\kappa_2\ge\kappa_3\ge\cdots\ge\kappa_p$. Lemma 1: $c_1\log n<p<c_2\log n$. Lemma 2: if $q$ is a prime with $\tfrac12p<q\le p$, then $\kappa_q=1$. Lemma 3: if $q$ is a prime with $2\sqrt p<q<4\sqrt p$, then $\kappa_q=2$. The paper says they are contained substantially in Ramanujan's 1915 paper and proves them on pp. 132–133 for completeness.

Read depth

Claims checked: the definition, the Theorem, the counting consequence, the three lemmas and Ingham's statement were read clause by clause on the page images of the print. The proof was read for its structure and is not reconstructed or independently reviewed here.

Source. P. Erdős, On highly composite numbers, J. London Math. Soc. 19 (1944), 130–133, doi:10.1112/jlms/19.75_part_3.130; the edition read is named on the source card.

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 $[1,x]$. The counting consequence gives $Q(x)>(\log x)^{1+c}$ for one unspecified $c>0$, so the bound asked for holds for the exponents $k\le1+c$; it does not decide the question for larger $k$, which the paper leaves open (question, p. 130).