Erdos 1979 propinquity divisors

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theorem_p304: Erdős and Hall's theorem that, for fixed eps > 0, with eta(x) equal to 3 to the power -(1+eps) sqrt(2 log log x . log log log log x), only o(x) integers n < x have divisors d < d' < d(1 + eta(x)(log d)^{1 - log 3}), with the equivalent density-zero form and the three remarks printed after it.

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P. Erdős, R. R. Hall: The propinquity of divisors, Bull. London Math. Soc. 11 (1979) no. 3, 304--307 (MR 81m:10102; Zentralblatt 421.10027).

Erdos and Hall sharpen Erdos's 1964 statement that integers with two very close divisors have density zero, making it precise particularly for small divisors. Their Theorem (p. 304) fixes eps > 0, sets eta(x) = 3^{-(1+eps) sqrt(2 log log x . log log log log x)} and Theta(x,d) = eta(x) (log d)^{1 - log 3}, and proves that the number of n < x having divisors d < d' < d(1 + Theta(x,d)) is o(x); equivalently the set of n with divisors d < d' < d(1 + Theta(n,d)) has asymptotic density zero. The proof (pp. 304--307) reduces to coprime pairs, treats divisors d > x^delta first with weights y^{Omega(n)}, and handles the rest with weights depending on the number of prime factors of n up to d, controlled by a Lemma (p. 307) drawn from Theorem VI of Erdos's 1946 paper on additive functions. Remarks (p. 304) note that the theorem is false if Theta depends on d alone, unless trivially Theta <= 1/d, since multiples of d(d+1) have positive density, and that it is unclear whether eta(x) is the most slowly decreasing function that works. The introduction also records that Erdos's 1964 claim that the density is 1 for exponent beta < log 3 - 1 "has had to be withdrawn" (p. 304).

Source: https://users.renyi.hu/~p_erdos/1979-26.pdf. No notice is printed in the file (pp. 304--305 and 306--307 read); the society's journals page (https://www.lms.ac.uk/publications/jlms, read 2026-10-02) prints "© Copyright London Mathematical Society 2026", names Wiley as the publisher that handles rights and permissions through Wiley Online Library, and states that the Bulletin shares the Journal's hybrid open-access arrangement with no blanket license, and Wiley Online Library could not be read on 2026-10-02; every other right reserved.

Bears on. #144: the Theorem (p. 304) is a density-zero result for divisors far closer than d < d' < 2d and proves nothing toward the problem's density-one statement; the introduction (p. 304) records the withdrawal of Erdos's 1964 claim of density 1 for divisors d < d' < d(1 + (log n)^{-beta}) with beta < log 3 - 1.

Results.

• Theorem (p. 304, unnumbered), with its density-zero form and Remarks (i)--(iii): for fixed eps > 0, only o(x) integers n < x have divisors d < d' < d(1 + Theta(x,d)).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.