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Source. Theorem 1, p. 3, of P. Erdős, S. W. Graham, A. Ivić and C. Pomerance, On the number of divisors of n!, Analytic Number Theory (Progress in Mathematics), Birkhäuser Boston (1996), 337--355, doi:10.1007/978-1-4612-4086-0_19, read in the authors' manuscript named on the source card; pages here are that manuscript's printed pages 1--16, and the published pagination was not compared.

Statement

Notation (pp. 1--3). $d(m)$ is the number of positive divisors of $m$, and $[x]$ is the integer part of $x$. The constants are defined by display (3) on p. 3:

$$c_k=\int_1^\infty\frac{\log([t]+1)}{t^2}\log^k t\,dt\qquad(k\ge0).$$

Theorem 1 (p. 3). "For any fixed integer $K\ge0$ and $c_k$ given by (3) we have

$$d(n!)=\exp\Bigl\{\frac{n}{\log n}\sum_{k=0}^{K}\frac{c_k}{\log^k n}+O\Bigl(\frac{n}{\log^{K+2}n}\Bigr)\Bigr\}."$$

In particular (p. 3), $c_0=\sum_{k\ge2}\frac{\log k}{k(k-1)}\approx1.25775$, so $\log d(n!)\sim c_0\,n/\log n$. With $m=n!$ and Stirling's formula the paper restates the case $K=0$ on p. 3 as

$$\log d(m)=\frac{c_0\log m}{(\log\log m)^2}\Bigl(1+O\Bigl(\frac{\log\log\log m}{\log\log m}\Bigr)\Bigr),$$

to be compared with Wigert's bound $\log d(m)\le\log2\,\log m/\log\log m+O(\log m/(\log\log m)^2)$ for all $m$ (display (1), p. 1).

Read depth. Claims checked: the statement, display (3) and the value of $c_0$ were read clause by clause on the page images on 2026-10-08; the proof on pp. 2--3 was read for structure only. Nothing here is independently reviewed.

Proof sketch

Pp. 2--3. Write $n!=\prod_{p\le n}p^{w_p(n)}$ with $w_p(n)=\sum_{j\ge1}[n/p^j]$, so $\log d(n!)=\sum_{p\le n}\log(w_p(n)+1)$. The primes $p\le n^{3/4}$ contribute $O(n^{3/4})$, since $w_p(n)<n/(p-1)$. For $p>n^{3/4}$ one has $w_p(n)=[n/p]$, and the prime number theorem with error $O(xe^{-\sqrt{\log x}})$ turns the sum into $\int_{n^{3/4}}^{n}\log([n/x]+1)\,dx/\log x$ plus $O(ne^{-\frac12\sqrt{\log n}})$. Substituting $x=n/t$ and expanding $1/\log(n/t)$ in powers of $\log t/\log n$ gives the stated expansion.

Dependencies

The prime number theorem with the classical error term (the paper cites Davenport and Ivić's book); no other result of the paper.

Bears on

• Problem 420: the paper remarks on p. 5 that Theorem 1 immediately gives that the average order of $K(n)$, the least $K$ with $d((n+K)!)\ge2d(n!)$, is of the order of $\log n$. This places the problem's shift $\log n$ at the typical scale for doubling; it does not bear on any of the problem's questions directly.