Davenport 1951 sequences positive integers

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main_theorem: Gives an elementary proof that the integers divisible by some term of an infinite increasing sequence have lower density and logarithmic density both equal to the limit A of the finite inclusion-exclusion densities.

remark_p19: Shows that when the reciprocals of the sequence have a convergent sum, the integers divisible by some term have a natural density equal to the limit A of the finite inclusion-exclusion densities.

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H. Davenport, P. Erdős: On sequences of positive integers, J. Indian Math. Soc. (N.S.) 15 (1951), 19--24 (MR 13,326c; Zentralblatt 43,49). No DOI has been established for the paper, and no check of the MR and Zentralblatt identifiers or of the scan URL against publisher metadata is recorded. Its reference [2] dates the authors' same-titled predecessor in Acta Arithmetica 2, 147--151, to 1937, where the repository's card for that paper uses its 1936 publication identity; the two same-titled papers should not be conflated.

The source copy is a scan; the optical text is noisy but readable. The note revisits the authors' 1936 theorem that, for the sequence b_1, b_2, ... of all integers divisible by some a_j, the number A = lim_m A(a_1, ..., a_m) is both the lower natural density and the logarithmic density of the b sequence. The 1936 proof used Dirichlet series and a Hardy-Littlewood Tauberian theorem; the stated object here is to replace it with a direct and elementary argument. Writing d, D for the lower and upper natural densities and delta, Delta for the lower and upper logarithmic densities (p. 20 introduces each pair as "upper and lower", but its chain and its reduction to (4) use d and delta as the lower ones), they note the general chain d <= delta <= Delta <= D and the easy inequality d >= A, so the whole theorem reduces to proving that the upper logarithmic density Delta is at most A, which is what the elementary argument supplies. They also record the simple case where sum 1/a_n converges, in which A is the ordinary density outright, and recall Besicovitch's example showing the b sequence need not have a natural density. The corpus's claim page for problem 26 starts from the convergent case, and the elementary theorem is the second publication behind the zero-class case of problem 486. Problem 1217's references also list the paper, which has no divisibility-chain result: the chain theorem that Erdős, Sárközy and Szemerédi (1966) credit to Davenport and Erdős, citing the Acta paper with this one as "see also", is Theorem 2 of the 1936 paper, whose proof uses the logarithmic-density theorem reproved here.

The copy read for this card is the six-page scan at the source URL below, whose PDF pp. 1--6 are printed pp. 19--24; the footnote on p. 19 reads "Received January 31, 1951." Read status: claims checked; the hypotheses, conclusions and equation locators below were checked against the scan's page images; the proof mechanism was traced for comparison with the 1936 argument, but no independent proof verification is recorded.

Statement and locators. Equation (1), p. 19, gives A_m, the density of the union of the multiples of a_1, ..., a_m, by inclusion-exclusion in the least common multiples, and equation (2), same page, defines the increasing limit A. The theorem, restated on p. 20 and proved through p. 23, is that the set B of multiples has lower natural density A and logarithmic density lim (log x)^{-1} sum_{b <= x, b in B} 1/b = A. Pages 19--20 isolate the easier stronger case: if sum 1/a_j converges then B has natural density A, by bounding the omitted tail with sum_{j > m} 1/a_j.

Direct proof and comparison with 1936. Since B contains every finite union, the lower natural density is at least A, so by the density chain it suffices to prove the upper logarithmic bound (4), limsup beta(x)/log x <= A, where (3) defines beta(x) = sum_{b <= x} 1/b (pp. 20--21). The replacement for the 1936 Tauberian argument is a finite-prime approximation: for the first k primes let Pi_k be the reciprocal sum over the integers supported on those primes (equation (5), p. 21) and B_k the normalized reciprocal mass of the members of B in that semigroup (equation (6)); inclusion-exclusion within the semigroup gives B_k = A(a'_1, a'_2, ...), where the a'j are the generators supported on those primes (equation (7), p. 21), and the truncation argument of equation (8), p. 22, shows B_k increases to A. For fixed k the b <= x are split into those divisible by a k-smooth generator, of logarithmic density B_k (equation (9), p. 22), and the remainder, whose reciprocal mass for p_h <= x < p{h+1} is at most Pi_h (B_h - B_k) <= C (B_h - B_k) log x by equations (10)--(12) and the bound Pi_h < C log p_h (pp. 22--23); letting x and then k tend to infinity proves (4) on p. 23. The 1936 proof instead writes the indicator Dirichlet series as F(s) = zeta(s) A(s), proves monotonicity of its normalized finite approximants from divisibility-upward closure, obtains F(s) ~ A/(s-1), and invokes Hardy and Littlewood's Tauberian theorem. The 1951 argument is elementary in that analytic sense but retains the same structural reliance on a union of sets of multiples.

Source: https://users.renyi.hu/~p_erdos/1951-07.pdf. No notice is printed in the copy read (pp. 1--2 and 5--6 read); the hosting archive's site footer (https://users.renyi.hu/~p_erdos/), "(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only.", speaks for the site, not the paper; the paper has no DOI and so no Crossref record, and the journal's site (informaticsjournals.co.in) could not be read; the term is unstated.

Bears on.

• #26: the convergent case (pp. 19--20) gives the multiples of a sequence with convergent reciprocal sum the natural density A; the problem's claim page crediting this paper starts from it, and the further steps, that A is then below one when every a_j >= 2 and so that no shift of such a set has almost all integers as multiples, are the claim page's own.
• #486: the main theorem (pp. 20--23) gives the set of all multiples a logarithmic density; the problem's claim page for the case where every residue set is the zero class passes from it to the problem's set of non-proper-multiples through Behrend's bound, a step the paper does not take. Other residue choices are not covered.
• #25: context only. The paper concerns sets of multiples, while the problem sieves one residue class per modulus.
• #1217: context only. The problem's references list the paper, which has no divisibility-chain result; the chain theorem behind the citation is Theorem 2 of the 1936 paper, whose proof uses the logarithmic-density theorem reproved here.

Results.

• Main theorem (unnumbered, stated p. 20, proved pp. 20--23): the set of multiples of an infinite increasing sequence has lower density A and logarithmic density A, proved elementarily without Dirichlet series or Tauberian theorems.
• Remark, p. 19 (convergent case, proved pp. 19--20): if sum 1/a_n converges, the set of multiples has natural density A.

The closing remark (pp. 23--24) that densities essentially stronger than the logarithmic one need not exist is recorded on the main theorem's page. The note added May 1951 (p. 24), which states that for a given positive integer k the b_i with b_{i+1} - b_i = k have a logarithmic density, and a natural density when the whole b sequence has one, and indicates the method in one sentence without a proof, has no page of its own.

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