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Some of Erd\H os' unconventional problems in number theory, thirty-four years later

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Pith's one-line read Tenenbaum surveys progress on Erdős' problems about divisors, sets of multiples, and prime factors, and proves that the distribution of τ+(n)/τ(n) is continuous at 1.

desk verdict A genuine but small new result inside a useful survey; the proof of Theorem 1 is too sketchy to verify from the page alone, but the paper deserves referee time. read the letter →

arxiv 1908.00488 v1 pith:QVT35H4T submitted 2019-08-01 math.NT

Gérald Tenenbaum This is my paper · ORCID
classification math.NT
keywords numberproblemssometheoryaccountgivehistoricalincluding
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper is a historical survey by Gérald Tenenbaum, an expert in analytic number theory. It goes through a list of problems that Paul Erdős posed in 1979 and explains what has been proved about them in the following thirty-four years. The main topics are the distribution of divisors of integers, sets of multiples, Behrend sequences, and the k-th prime factor or k-th divisor of a random integer. In each case Tenenbaum states the original problem, the current best results, and the references. The survey also contains two short proofs: one showing that a certain distribution function is continuous at the point 1, and one giving a direct proof that a sequence with logarithmic density 1 has a set of multiples with natural density 1. The first of these is presented as answering an open question about discontinuity points. The paper is written for specialists; it assumes fluency with the notation of probabilistic number theory. It does not include code or data, because it is a mathematics survey. Erdős' problems are about how divisors and prime factors are arranged for a typical integer n. For example, one question asks whether almost every integer has two divisors that are close together, with one less than twice the other. Tenenbaum explains how this was settled by work of Erdős and Hall and by Maier and Tenenbaum. He then follows the descendants of the problem: the Erdős-Hooley Delta function, the propinquity functions, the distribution of the k-th smallest divisor, and the density of integers with a divisor in a short interval. There are also sections on Behrend sequences, which are sets of multiples that contain almost all integers, and on the relative sizes of the largest prime factors of n and n+1.
Extended reading notes

Core claim

The paper's strongest mathematical assertion is Theorem 1: 'The distribution function ν is continuous at z = 1.' Here ν(z) is the limiting distribution of τ+(n)/τ(n), where τ+(n) counts the number of dyadic intervals that contain at least one divisor of n. If true, this answers, for the endpoint z=1, a question about the discontinuity points of ν that the author reports as open.

Load-bearing premise

The proof of Theorem 1 relies on Lemma 48.1 from Hall and Tenenbaum's book Divisors (1988). The lemma asserts that the discrepancy of the sequence {(log m)/log 2 : m | (n/nε)} is at most ε on a subsequence of lower density 1 - ε/3. The proof also uses Theorem 51 of the same book, which provides two close divisors on a set of density close to 1. If either of these cited results is false or is misapplied here, the continuity conclusion would not follow. These lemmas are not restated in the paper, only cited.

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Summary. This paper is a historical survey of Erdős's 1979 article [24] on unconventional problems in number theory, thirty-four years after its publication. The author updates the status of each problem, describes descendants and posterity, and includes two original results: Theorem 1 proves that the limiting distribution ν(z) of τ_+(n)/τ(n) is continuous at z = 1, and Theorem 2 gives a direct proof that an integer sequence with logarithmic density 1 has a set of multiples with natural density 1. The survey covers close divisors, the Erdős–Hooley Delta function, Behrend sequences, divisor statistics, and problems on largest prime factors of consecutive integers, with extensive references.

Significance. If correct, Theorem 1 answers the endpoint part of an open problem raised in the paper, namely the determination of discontinuity points of ν. Theorem 2 provides an elementary self-contained proof of a statement that follows from the Davenport–Erdős theorem, and the proof is interesting in its own right. As a survey, the paper is authoritative and unusually informative: it is written by a leading contributor to the subject and includes many recent results with precise statements. The author is careful to point out limitations, notably in footnote 3, where he notes that Theorem 1 does not yield a new proof of Erdős's conjecture (9) because a refinement of (9) was used. The survey statements appear accurate, and the proofs of the two theorems are plausible; Theorem 2's proof is self-contained.

minor comments (5)
  1. [Proof of Theorem 1] The proof relies on Theorem 51 and Lemma 48.1 of [46] without stating them; please include their precise statements or at least a clear description so that the proof can be checked without the book. Also fix the notation '2εd' to '2^ε d' and 'T ε' to 'T_ε'.
  2. [Proof of Theorem 1] In the same proof, specify that the discrepancy in Lemma 48.1 refers to the fractional parts {(log m)/log 2} modulo 1, since the argument depends on distances to integers.
  3. [Proof of Theorem 1] The assertion 'τ(n_ε) ≤ log T_ε holds on a sequence of lower density 1 − ε/3' is stated without proof or reference; please add a brief justification or a citation to a standard result on smooth parts of integers.
  4. [Footnote 3] Footnote 3 correctly notes that Theorem 1 does not provide a new proof of (9) because a refinement of (9) was used; this circularity is acknowledged, but it would help to also mention this dependency near the statement of Theorem 1 so readers are not misled.
  5. [Throughout] There are numerous OCR/typographical errors (e.g., '3 ω (n)' for '3^{ω(n)}', 'greaterorequalslant' for '≥', 'd, d, d' for the three density symbols); a careful proofreading pass would improve readability.
Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is a survey and relies on standard theorems of analytic number theory, including the Davenport-Erdős theorem, Mertens' theorem, the Erdős-Kac theorem, and results from Hall-Tenenbaum's book. No free parameters or invented entities are introduced.

assumptions (4)
  • standard math Davenport-Erdős theorem: for any integer sequence A, the natural density and logarithmic density of the set of multiples M(A) are equal.
    Invoked in the discussion of Behrend sequences and in the proof of Theorem 2 to deduce that dM(A)=1 from δA=1 (see Theorem 2 proof and following paragraph).
  • standard math Existence and basic properties of the distribution function ν of τ+(n)/τ(n), established in Hall-Tenenbaum's Divisors (1988).
    Used in Theorem 1 to define ν and to apply the estimates for ν and the divisor discrepancy lemmas.
  • standard math Mertens' theorem on the product over primes ≤ y of (1 - 1/p).
    Used in the proof of Theorem 2 to estimate the sum over integers with no small prime factors.
  • standard math Erdős-Kac theorem on the Gaussian distribution of ω(n).
    Used in the heuristic discussion of the independence of prime factors leading to conjecture (1).

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Pith. "Pith review of Some of Erd\H os' unconventional problems in number theory, thirty-four years later." pith.science (2026). https://pith.science/paper/QVT35H4T

@misc{pith2026190800488,
  author       = {Pith},
  title        = {Pith review of: Some of Erd\H os' unconventional problems in number theory, thirty-four years later},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QVT35H4T}},
  note         = {Machine review of arXiv:1908.00488}
}
read the original abstract

We give an historical account, including recent progress, on some problems of Erd\H os in number theory.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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