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Source. Theorem 6, p. 14, 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

Definitions as on the page of Theorem 5: $D(n)=d(n!)-d((n-1)!)$, and $n$ is a champ if $D(n)>D(m)$ for all natural $m<n$.

Theorem 6 (p. 14). "Assuming the Riemann Hypothesis, the set of champs has asymptotic density zero."

The authors state just before it (p. 14) that they conjecture the density statement unconditionally and cannot prove it. After the proof (p. 15) they note that a conjecture of Cramér would give at most $O(x\log\log x/\log x)$ champs up to $x$, against the lower bound $\gg x/\log x$ from the primes, and that the upper density of the set of champs is less than $1$ (by the method of Erdős and Pomerance on the largest prime factors of $n$ and $n+1$).

Read depth. Claims checked: the statement was read clause by clause on the page image on 2026-10-08, and the proof on pp. 14--15 was followed. Nothing here is independently reviewed.

Proof sketch

Pp. 14--15. Unconditionally, for large $n$: if $P(n)\le n/\log^3n$ and the interval $(n-\frac13\log^3n,n]$ contains a prime, then $n$ is not a champ. Indeed, for a champ with $P(n)\le n/\log^3n$, Theorem 2 makes $D(n)$ at most about $\log^{-3}n$ times $d((n-1)!)$, and summing $D(k)<D(n)$ over $m<k<n$ with $m=[n-\frac13\log^3n]$ gives $d((n-1)!)<2d(m!)$, while a prime $p$ in $(m,n]$ would double the divisor count. The $n$ with $P(n)\le n/\log^3n$ have density $1$ unconditionally, and under the Riemann Hypothesis a theorem of Selberg gives that the intervals $(n-\frac13\log^3n,n]$ contain a prime for a set of $n$ of density $1$.

Dependencies

Theorem 2 and Selberg's theorem on the normal density of primes in short intervals under the Riemann Hypothesis.

Bears on

No problem page in the corpus concerns the champs of $D(n)$.