Erdos 1974 distribution numbers form sigma n n

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question_p59: Erdős records that the distribution functions of sigma(n)/n and phi(n)/n are purely singular, and states that it is not known whether the derivative of the distribution function of sigma(n)/n can take any value other than 0.

theorem_p60: Erdős's unnumbered Theorem: an absolute constant c_1 bounds the number of integers n up to x with a at most sigma(n)/n below a + 1/t by c_1 x / log t when x exceeds t, a bound best possible apart from c_1.

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P. Erdos, On the distribution of numbers of the form sigma(n)/n and on some related questions. Pacific Journal of Mathematics 52 (1974), 59-65. The copy read for this card prints only the journal masthead; the publisher's article page states "© Copyright 1974 Pacific Journal of Mathematics. All rights reserved." with no open-access or license statement (https://msp.org/pjm/1974/52-1/p08.xhtml, read 2026-10-02), every other right reserved.

Erdos proves a single Theorem (p. 60): there is an absolute constant c_1 such that for x > t (the print reads "for 0, x > t") the count F(x; a, a + 1/t) of integers n <= x with a <= sigma(n)/n < a + 1/t is less than c_1 x / log t, best possible apart from c_1, which sharpens a result of Tyan (Tjan). He notes, without proof, the slightly stronger form (1') with a(1 + 1/t) in place of a + 1/t, and the deduction (2), following Diamond, that F(x; 1, a) = x g(a) + o(x / log x), which sharpens Feinleib and has a best-possible error term; he says his earlier asymptotic (3) for F(x; 1, 1 + eps) as eps -> 0 implies that (1), if true, is best possible. The method mirrors his count of primitive abundant numbers: write each qualifying b as u v w by prime-factor size, discard sparse classes such as those divisible by a prime power p^alpha with alpha > 1 that exceeds (log t)^2, and bound what remains. The introductory survey is the part that bears on problem 50: it credits Schoenberg (1928) with the continuous distribution function for phi(n)/n, Behrend, Chowla and Davenport with the same for sigma(n)/n, and Erdos's own 1939 work with proving both distribution functions purely singular, so their derivatives vanish almost everywhere. For the distribution function g of sigma(n)/n he then states that it is not known whether the derivative can take any value other than 0, that he does not know whether the right or left derivative can take any value other than 0 or infinity, that the right derivative at c = sigma(n)/n is infinite, and that a dense set of c has no one-sided derivatives; the positive-derivative question is thus posed for sigma(n)/n, the analogue of problem 50's phi(n)/n, and not settled here. He remarks that the Theorem also holds with Euler's phi in place of sigma, with slightly simpler proofs.

Source: https://msp.org/pjm/1974/52-1/pjm-v52-n1-p08-p.pdf.

Bears on. #50: the problem asks whether the distribution function of phi(n)/n ever has a positive derivative; the introduction (pp. 59-60) asks the analogous question for sigma(n)/n, whether the derivative of its distribution function can take any value other than 0, and records that both distribution functions are purely singular, and the Theorem in its phi form, which the paper asserts without proof, would bound the increase of the distribution function of phi(n)/n over an interval of length 1/t by c_1/log t. Neither settles the problem.

Results. the Theorem (p. 60, unnumbered), with the remarks (1'), (2), (3) and those of pp. 63-64 summarized on its page; the derivative question (pp. 59-60, unnumbered).

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