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Source. Lemma 3, p. 8, 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

Lemma 3 (p. 8). "Let $x$ be a sufficiently large positive real number, let $c=4/9$, and $\delta=1/10000$. Then the number of primes $p$ such that $p$ divides some $m$ in the interval $(x,x+x^c]$ and $p>x^{1-c+\delta}$ is $\gg x^c$."

The proof ends (p. 12) with the weighted form $\sum_{p>x^{5/9+\delta}}N(p)\log p>0.0002\,x^{4/9}\log x$ for large $x$, where $N(p)$ is the number of multiples of $p$ in $(x,x+x^{4/9}]$.

Read depth. Claims checked: the statement was read clause by clause on the page image on 2026-10-08; the proof on pp. 9--12 was read for structure only, and the numerical claim $4/9+I(4/9)<0.9998$ on p. 12 was not recomputed. Nothing here is independently reviewed.

Proof sketch

Pp. 9--12, an adaptation of Ramachandra's argument for large prime factors of integers in short intervals. Chebyshev's identity $\sum_{x<m\le x+x^c}\log m=\sum_d\Lambda(d)N(d)$ shows that the primes above $x^c$, with their powers, carry weight $(1-c)x^c\log x+O(x^c)$. The part of this weight from primes in $(x^c,x^{1-c+\delta}]$ is bounded above with Selberg's upper bound sieve, the error terms being handled by exponent pairs (the pairs $(1/2,1/2)$, $(1/6,2/3)$ and $(1/14,11/14)$ on three ranges). With $c=4/9$ and $\delta=10^{-4}$ that part falls short of the total by a positive multiple of $x^{4/9}\log x$, which the primes above $x^{5/9+\delta}$ must supply. Each such prime exceeds the length $x^{4/9}$ of the interval, so it divides at most one of its integers, and with $\log p\le\log(2x)$ the weighted bound gives the count.

Dependencies

Chebyshev's identity, Selberg's upper bound sieve as presented in Hooley's book, and Lemma 4.3 of Graham and Kolesnik's book on exponent pairs.

Bears on

• Problem 420: through Corollary 3, whose proof rests on this lemma; see that page for the relation.