Erdos 1964 applications probability analysis number theory

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item_1: Erdős's unpublished announcements that almost all integers up to n have divisors in every residue class mod m when m < 2^{(1-eps_1) log log n}, that about (1+eps)(log n)/log 2 random elements of an abelian group of order n almost always give every element as a 0-1 product, and an asymptotic for Pillai's count Q(n).

item_2: Erdős's announcement, without proof and qualified by "Unless I made a mistake", that for every eta > 0 almost all n have two divisors with d_1 < d_2 < d_1(1 + (e/3)^{(1-eta) log log n}), while the integers with 1+eta in place of 1-eta have density zero.

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P. Erdős: On some applications of probability to analysis and number theory, J. London Math. Soc. 39 (1964), 692--696 MR 30 #1997; Zentralblatt 125,86.

This is a short survey of Erdős's own probabilistic results in analysis and number theory: gap power series that converge uniformly in the unit disc with divergent coefficient sums, singular radii of power series (with Rényi), a negative answer for p > 2 to Zygmund's L^p analog of Wiener's gap theorem, everywhere divergence of randomly signed series (with Dvoretzky) when |a_k| >= c_k for a monotone sequence c_k tending to zero with limsup (c_1^2 + ... + c_k^2)/log(1/c_k) > 0, and a central limit theorem for lacunary trigonometric sums with all coefficients 1 under the weakened gap condition n_{k+1} > n_k(1 + c_k/k^{1/2}) with c_k tending to infinity, proved by the method of moments. Unpublished number-theoretic announcements include that almost all u <= n have divisors in every residue class mod m when m < 2^{(1-eps)log log n} (with the complementary statement above that threshold), via a result that a random set of about (1+eps)(log n)/log 2 elements of an n-element abelian group has all subset sums covering the group, and the resulting asymptotic for Pillai's counting function Q(n). The survey also recalls the published Erdős-Rényi probabilistic proof that some sequence with a_k < k^{2+eps} has a bounded number of representations as sums of two terms. Bearing on problem 144, Erdős states he had earlier proved that the density of integers with two divisors d_1 < d_2 < 2 d_1 exists, and announces here ("Unless I made a mistake", p. 696) that this density is 1, in the sharper form that for every eta > 0 the density is 1 for d_1 < d_2 < d_1(1 + (e/3)^{(1-eta) log log n}), that is, a gap (log n)^{-(1-eta)(log 3 - 1)}, and 0 with 1+eta in place of 1-eta (printed pp. 695--696, displays (8) and (9)). Erdős and Hall withdrew the density-one claim (8) in 1979 while proving (9) in a quantitative form, and Maier and Tenenbaum proved (8), and with it the density-one statement, in 1984.

Source: https://users.renyi.hu/~p_erdos/1964-15.pdf. No notice is printed in the copy read (pp. 692--693 and 695--696 read); the society's journals page (https://www.lms.ac.uk/publications/jlms) prints "© Copyright London Mathematical Society 2026", names Wiley as the publisher that handles rights and permissions through Wiley Online Library, and names no blanket license for the hybrid open-access journal, and Wiley Online Library could not be read; every other right reserved.

Bears on. #144: item 2 announces without proof, qualified by "Unless I made a mistake" (p. 696), that the density of integers with two divisors d_1 < d_2 < 2 d_1 is 1, via the sharper display (8); the paper proves nothing toward the problem.

Result pages. Claims checked on the page images of the print: item 1 (p. 695: divisors in residue classes, random subset products, Pillai's Q(n)) and item 2 (pp. 695--696: displays (8) and (9)).

Results to transcribe.

• Divisor density announcement (pp. 695--696, displays (8) and (9) on p. 696): The density of integers having two divisors d_1 < d_2 < 2 d_1 is claimed to be 1, in the sharper form d_1 < d_2 < d_1(1 + (e/3)^{(1-eta) log log n}), a gap of (log n)^{-(1-eta)(log 3 - 1)}, with density 0 when 1+eta replaces 1-eta; the claim (8) was withdrawn in 1979 and proved by Maier and Tenenbaum in 1984.
• Divisors in residue classes (p. 695): For n > n_0(eps_1, eps_2) and m < 2^{(1-eps_1) log log n} all but eps_2 n integers u <= n have divisors in every residue class mod m (the print has 1 <= u <= m); for m > 2^{(1+eps_1) log log n} the print says that fewer than eps_2 n integers u < n "have a divisor in any given residue class mod m".
• Random subset sums in abelian groups (p. 695): In an abelian group of n elements, for almost all choices of k about (1+eps)(log n)/log 2 elements, every element is a 0-1 combination of them.
• Pillai's function (p. 695): Q(n), the count of m <= n with no divisor of the form p(kp+1), satisfies Q(n) = (1+o(1)) e^{-gamma} n/(log 2 * log log n).
• Lacunary CLT (p. 694, display (7)): The Gaussian limit law for sums of cos/sin at frequencies n_k holds under n_{k+1} > n_k(1 + c_k/k^{1/2}) with c_k tending to infinity, when all coefficients are 1.
• Erdős-Rényi sequence with bounded representation counts (p. 695): For every eps there is a sequence with a_k < k^{2+eps} for which the number of representations n = a_i + a_j is bounded.

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