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Statement

Setting (pp. 695--696, item 2 of the closing list of unpublished number-theoretic results). Erdős recalls that he had earlier proved (the paper's reference [13], "Density of some sequences of integers", 1948) that the density of the integers $n$ having two divisors $d_1,d_2$ with $d_1<d_2<2d_1$ exists, without being able to show that it equals $1$.

Announcement (p. 696). Introduced by the words "Unless I made a mistake I proved this recently" (p. 696), Erdős states that for every $\eta>0$:

• the integers $n$ having two divisors $d_1,d_2$ with

$$d_1<d_2<d_1\Bigl(1+\bigl(\tfrac e3\bigr)^{(1-\eta)\log\log n}\Bigr) \tag{8}$$

have density $1$;

• the integers $n$ having two divisors $d_1,d_2$ with

$$d_1<d_2<d_1\Bigl(1+\bigl(\tfrac e3\bigr)^{(1+\eta)\log\log n}\Bigr) \tag{9}$$

have density $0$.

Here $n$ in the exponent is the integer whose divisors are counted. Since $(e/3)^{c\log\log n}=(\log n)^{-c(\log3-1)}$, the bounds read $d_2/d_1<1+(\log n)^{-(1\mp\eta)(\log3-1)}$. Because $e/3<1$, the bound in (8) tends to $1$, so (8) would give density $1$ for $d_1<d_2<2d_1$, the statement the paper says it had set out to prove.

No proof is given. The paper adds only that the proof of (8) is comparatively simple and does not require probabilistic methods.

Later history. Erdős and Hall (1979) recorded that the claim (8) was withdrawn and proved (9) in a sharper form (source card); Maier and Tenenbaum (1984) proved (8) for every $\eta>0$, and with it density $1$ for $d_1<d_2<2d_1$ (Theorem 1).

Source. P. Erdős, On some applications of probability to analysis and number theory, J. London Math. Soc. 39 (1964), 692--696; item 2 runs from the foot of p. 695 to p. 696, with displays (8) and (9) on p. 696. The edition read is named on the source card.

Read depth. Claims checked: the statement was read clause by clause on the page images of the print. The paper gives no proof.

Proof pointer

None in this paper. For (8) see Maier and Tenenbaum, Invent. Math. 76 (1984), 121--128; for (9) in sharper form see Erdős and Hall, Bull. London Math. Soc. 11 (1979), 304--307.

Dependencies

None in the paper. The existence of the density is cited from the paper's reference [13].

Bears on

• Problem 144: the paper announces, without proof and with the reservation quoted above, that the density of integers with two divisors $d_1<d_2<2d_1$ is $1$, through the sharper (8); it proves nothing toward the problem.