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Source. G. Tenenbaum, Some of Erdős' unconventional problems in number theory, thirty-four years later, in L. Lovász, I. Z. Ruzsa and V. T. Sós (eds), Erdős Centennial, Bolyai Society Mathematical Studies 25 (2013), 651--681. Labels and pages here are those of the author's version identified on the source card, paginated 1--22, which carries some corrections with respect to the published chapter; the published chapter was not read. Statement (1) is displayed on p. 2, inside the quotation from Erdős's 1979 article; its status is reported on p. 4.

Read depth. Claims checked: the statement and the status report were read clause by clause on the printed pages. The survey proves none of this; it reports results of other papers.

Statement

Statement (1) (p. 2, from the passage of Erdős that the survey quotes). Erdős claimed that almost all integers $n$ have two divisors with

$$d_1<d_2<d_1\bigl\{1+(\mathrm e/3)^{(1-\eta)\log\log n}\bigr\},$$

and that this is best possible, in that it fails when $1-\eta$ is replaced by $1+\eta$. The quoted text adds that Erdős and Hall confirmed the second assertion but could not prove (1). The range of $\eta$ is not printed; the statement is read for each fixed $\eta$ with $0<\eta<1$.

Since $(\mathrm e/3)^{\log\log n}=(\log n)^{1-\log 3}$, the survey's heuristic (pp. 3--4) is that the smallest of the $3^{\omega(n)}$ numbers $\log(d'/d)$ over pairs of divisors should be of size $(\log n)^{1-\log3+o(1)}$ for almost all $n$.

Status (p. 4, quoted). "This conjecture, which is now a theorem, due to Erdős–Hall [27] for the lower bound and to Maier–Tenenbaum [55] for the upper bound". Here [27] is P. Erdős and R. R. Hall, The propinquity of divisors, Bull. London Math. Soc. 11 (1979), 304--307 (card), and [55] is H. Maier and G. Tenenbaum, On the set of divisors of an integer, Invent. Math. 76 (1984), 121--128 (card).

Related estimates reported (p. 4). With $\mathcal E=\{m=dd' : d<d'<2d\}$ as in (5) (p. 3) and $\mathcal M(\mathcal E)$ its set of multiples, Stef's thesis gives, for the number $R_x$ of integers up to $x$ outside $\mathcal M(\mathcal E)$,

$$x/(\log x)^{\beta+o(1)}\ll R_x\ll x\,\mathrm e^{-c\sqrt{\log\log x}}\qquad(8)$$

for some constant $c>0$, with $\beta=1-(1+\log\log3)/\log3\approx0.00415$, the best estimates known to the survey. Raouj, Stef and Tenenbaum prove that $E_1(n)=\min_j\log\{d_{j+1}(n)/d_j(n)\}$, over consecutive divisors, equals $(\log n)3^{-\omega(n)}(\log\log n)^{\vartheta_n}$ for almost all $n$, with $-5\le\vartheta_n\le10$.

Earlier in the survey (p. 3) Erdős's criterion (4) for a set of multiples to have a natural density is applied to $\mathcal E$, so the integers with two divisors $d<d'<2d$ have a natural density.

Proof pointer

None in the survey: the two halves are proved in [27] and [55].

Dependencies

Erdős and Hall 1979 and Maier and Tenenbaum 1984, as above.

Bears on

• Problem 144: for $0<\eta<1$ the factor $1+(\mathrm e/3)^{(1-\eta)\log\log n}$ tends to $1$, so the upper bound half of (1), credited to Maier and Tenenbaum, gives two divisors with $d_1<d_2<2d_1$ for almost all $n$; the survey reports that the density exists (p. 3) and that (1) is a theorem (p. 4), and (8) bounds the exceptions.