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Statement

Setting (pp. 116–119). $d(n)$ is the number of divisors of $n$, and $A$ is highly composite when every $M<A$ has $d(M)<d(A)$. $N$ is superior highly composite when for some real $\epsilon>0$ every integer $M$ satisfies $d(M)/M^\epsilon\le d(N)/N^\epsilon$ (p. 117). For $0<\epsilon<1$ the paper recalls from Ramanujan that a superior highly composite number $N=N_\epsilon$ attached to $\epsilon$ has the exponent $a_\lambda=\lfloor 1/(\lambda^\epsilon-1)\rfloor$ at each prime $\lambda$ (display (6)), and attaches to it

$$x=2^{1/\epsilon},\qquad x_k=x^{\log(1+1/k)/\log2}\quad(k\ge1)$$

(display (7)), so that $a_\lambda=k$ exactly when $x_{k+1}<\lambda\le x_k$ (display (8)). The benefit of an integer $M$ relative to $N=N_\epsilon$ (bénéfice, display (11), p. 118) is a sum of non-negative terms over the primes at which $M$ and $N$ differ; by Proposition 1 and display (12) it is the quantity

$$\operatorname{bén}M=\epsilon\log\frac MN-\log\frac{d(M)}{d(N)}\ \ge\ 0 .$$

Théorème 1 (p. 120). Let $A$ be a highly composite number and $N=N_\epsilon$ the superior highly composite number preceding $A$, and put $x=2^{1/\epsilon}$. There are two constants $\gamma>0$ and $C>0$ such that $\operatorname{bén}A\le Cx^{-\gamma}$.

The proof (p. 123) obtains $\gamma=\theta(1-\tau)/(\kappa+1)$, where $\theta=\log(3/2)/\log2$, $\tau=5/8$ is Ingham's exponent, and $\kappa$ is the exponent in Feldman's bound $\lvert v\theta-u\rvert>c_1/v^\kappa$ for all integers $u,v$ (p. 122). Before the theorem, Proposition 3 (p. 119) gives only $\operatorname{bén}A\le\epsilon+\log2$.

Proof pointer

Pp. 120–123. Between $N$ and $NP$, with $P$ the prime after $x$, the paper builds a family $M_h$, $-H\le h\le H$, by moving $\lvert h\rvert$ primes across $x_2$ in one direction and about $\lvert h\theta\rvert$ primes across $x$ in the other; displays (13) and (15) bound their benefits, and the values $\log d(M_h)$ are spaced by at most $\lVert v_n\theta\rVert\log2$, with $v_n$ a continued-fraction denominator of $\theta$. Feldman's refinement of Baker's theorem bounds that spacing by a negative power of $H$ (display (17)). Proposition 2, applied to $A$ and the two members of the family whose divisor counts bracket $d(A)$, bounds the benefit of $A$, and the choice of $H$ as a power of $x$ gives the theorem.

Dependencies

None in the corpus. Inputs named by the paper: Ingham's theorem $\pi(x+x^\tau)-\pi(x)\sim x^\tau/\log x$ for $5/8\le\tau\le1$ (display (4), p. 116); Feldman's lower bound for linear forms in logarithms (reference [3]); Ramanujan's properties of superior highly composite numbers (reference [8], §§ 32–34); the paper's Propositions 1 to 4 (pp. 118–120).

Read depth

Claims checked: the definitions, displays (6) to (8), (11), (12) and the theorem were read clause by clause on the page images of the print, and the value of $\gamma$ on p. 123. 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

No Erdős problem directly. The theorem is the input to Théorème 2 and Théorème 3, through which it reaches Problem 381.