Tenenbaum 2013 erdos unconventional problems number theory

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equation_1: Erdős's statement (1), quoted in the survey, that almost all n have two divisors with ratio below 1 + (e/3)^{(1-eta) log log n} and that the exponent is best possible, which Tenenbaum reports as a theorem of Erdős and Hall (lower bound) and Maier and Tenenbaum (upper bound).

equation_15: The survey's report that the proportion tau^+(n)/tau(n) of occupied dyadic intervals among the divisors of n has a limiting distribution nu with z/sqrt(log(2/z)) << nu(z) << z log(2/z) on (0, 1), so that Erdős's conjecture that tau^+(n)/tau(n) tends to 0 for almost all n is false.

equation_27: From the Davenport–Erdős formula for the lower density of a set of multiples and Behrend's inequality, the survey's consequence (27) that the reciprocal sum of a Behrend sequence diverges and that every tail of a Behrend sequence is again one.

estimate_p9: For the sum G(n) of the ratios d_i/d_{i+1} of consecutive divisors, the survey's lower bound G(n) > tau(n)/2p with p the least prime factor of n, so that G(n) tends to infinity for almost all n, and its two-sided bound for x log x minus the sum of G(n) up to x.

theorem_1: Tenenbaum's theorem that the distribution function nu of the proportion tau^+(n)/tau(n) of occupied dyadic intervals among the divisors of n is continuous at 1, so that nu(1 - eta) tends to 1 as eta tends to 0.

theorem_2: Tenenbaum's direct proof that if an integer sequence A has logarithmic density 1 then its set of multiples has natural density 1, without the Davenport–Erdős theorem that lower and logarithmic densities of a set of multiples agree.

theorem_3: The survey's statement of the Hall–Tenenbaum and Tenenbaum criterion for a block sequence with log(T_{j+1}/T_j) of order j^sigma (log j)^tau and log H_j of order (log j)^gamma / j^alpha to be a Behrend sequence, with critical exponent alpha_0(sigma), giving log 2 in Erdős's example.

theorem_4: The survey's statement of Hall and Tenenbaum's criterion for block sequences whose blocks are long: divergence of the sum of (log H_j / log T_j)^{delta_1} for some delta_1 > delta gives a Behrend sequence, and convergence for some delta_2 < delta rules one out.

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Tenenbaum, Gérald, Some of Erdős' unconventional problems in number theory, thirty-four years later. In L. Lovász, I. Z. Ruzsa and V. T. Sós (eds), Erdős Centennial, Bolyai Society Mathematical Studies 25 (2013), 651--681. The copy read for this card is the author's version from the author's page (https://tenenb.perso.math.cnrs.fr/PPP/, read 2026-10-02), which states no copyright, license or terms for its PDFs, and no notice is printed on pp. 1–2 or 21–22 of that copy; that version carries some corrections with respect to the published chapter (its footnote on p. 1), and the published chapter is not the copy read; the term is unstated.

This is a survey chapter from the Erdos Centennial volume in which Tenenbaum revisits the questions posed in Erdos's 1979 article on unconventional problems in number theory and reports their status thirty-four years later. It opens with Erdos's old conjecture that the set of integers n having two divisors with d_1 < d_2 < 2 d_1 has density 1, tracing its origin through primitive abundant numbers, the density of abundant integers, and the theory of sets of multiples (including Besicovitch's counterexample to natural density of sets of multiples). Erdos's 1979 text, quoted on pp. 1–2, says that he proved the density exists but never that it equals 1, and that he and Hall confirmed that the refined statement d_1 < d_2 < d_1 (1 + (e/3)^{(1-eta) log log n}) for almost all n is best possible while the statement itself stayed open. Tenenbaum records (p. 4) that this conjecture "is now a theorem, due to Erdős–Hall [27] for the lower bound and to Maier–Tenenbaum [55] for the upper bound", and gives Stef's bounds for the number of exceptional integers up to x. The method is expository: each problem is framed by Erdos's own quoted text, then updated with later theorems and remaining gaps. Two results are the survey's own, with proofs: Theorem 1 (p. 7), that the limiting distribution of tau^+(n)/tau(n) is continuous at 1, and Theorem 2 (p. 14), a direct proof that a sequence of logarithmic density 1 is a Behrend sequence; Theorems 3 and 4 (pp. 16--17) restate criteria for block Behrend sequences from other papers. For problem 26 the relevant passage is the one on Behrend sequences (sequences whose set of multiples has density 1): from the Davenport–Erdos formula (25) for the lower density of a set of multiples and the inequality (26) that it yields from Behrend's inequality for finite sequences, a divergent reciprocal sum (27) is a necessary condition for a sequence to be a Behrend sequence (p. 15).

Source: https://tenenb.perso.math.cnrs.fr/PPP/.

Edition. Labels and pages on this card and its result pages are those of the author's version named above, paginated 1--22; the published chapter was not read.

Read status: claims checked for every result page below, read clause by clause on the printed pages; Theorems 1 and 2, the survey's own, were read in outline for their proofs, and the other results are reported in the survey from other works and not proved there.

Bears on.

• #26: by (27) (p. 15) a Behrend sequence has a divergent reciprocal sum, so for an infinite A with convergent reciprocal sum no shift A + k (k >= 1) is a Behrend sequence, a negative answer for such A. The survey does not state the problem or draw this conclusion, and (27) as printed tacitly excludes 1 from the sequence.
• #144: the survey reports (pp. 3--4) that the integers with two divisors d_1 < d_2 < 2 d_1 have a density and that Erdős's sharper statement (1), which implies density 1, is a theorem of Erdős and Hall and of Maier and Tenenbaum; Theorem 1 implies the density-one statement again, which the survey says is no new proof.
• #448: the survey states that Erdős's conjecture tau^+(n)/tau(n) -> 0 for almost all n is wrong, and reports the bounds (15) (p. 7) for the limiting distribution nu of tau^+(n)/tau(n), whose upper bound nu(z) << z log(2/z) gives the negative answer.
• #673: pp. 8--9 give G(n) > tau(n)/2p with p the least prime factor of n, so G(n) -> infinity for almost all n, and report x log x - sum_{n <= x} G(n) of exact order x(log x)^{1-delta}/(log log x)^{3/2}, an asymptotic formula for the sum.
• #691: the survey quotes the problem (p. 13) and records partial answers: the necessary condition (27), the sufficient condition Theorem 2 (logarithmic density 1), and criteria for block sequences, Theorem 3, which settles Erdős's block example in its two-sided form with threshold log 2, and Theorem 4; it calls an effective general criterion seemingly hopeless (p. 15).

Results. Labels and pages are those of the author's version.

• Statement (1) (pp. 2, 4): almost all n have divisors d_1 < d_2 < d_1(1 + (e/3)^{(1-eta) log log n}), and this fails with 1 + eta in place of 1 - eta; reported as a theorem, due to Erdős and Hall for the lower bound and Maier and Tenenbaum for the upper bound (p. 4), with Stef's bounds (8) for the exceptions.
• Equation (15) (p. 7): tau^+(n)/tau(n) has a limiting distribution nu with z/sqrt(log(2/z)) << nu(z) << z log(2/z) for 0 < z < 1, reported from Hall and Tenenbaum's Divisors, Chapter 4.
• Theorem 1 (p. 7): nu is continuous at z = 1.
• Estimates for G(n) (pp. 8--9): G(n) > tau(n)/2p, and the two-sided bound for x log x - sum_{n <= x} G(n), the latter obtained from Theorem 3 of Erdős and Tenenbaum 1983 and Ford's estimate (16).
• Theorem 2 (p. 14): if the integer sequence A has logarithmic density 1, then its set of multiples has natural density 1.
• Equations (25)--(27) (p. 15): the Davenport–Erdős formula for the lower density of a set of multiples, Behrend's inequality, and the consequence that a Behrend sequence has a divergent reciprocal sum and Behrend tails.
• Theorem 3 (p. 16): the Behrend criterion with threshold alpha_0(sigma) for block sequences with short blocks, due to Hall and Tenenbaum (necessity) and Tenenbaum (sufficiency).
• Theorem 4 (p. 17): the criterion through the series of (log H_j / log T_j)^delta for block sequences with long blocks, due to Hall and Tenenbaum.

Surveyed without a result page here: Besicovitch's lim inf of the density of the multiples of (T, 2T] being 0, (2) on p. 2, and Erdős's improvement (3) on p. 3; the propinquity functions E_r(n), Raouj's theorem on the multiples of the intervals (d, 2d] over divisors d of n, and the Erdős–Hooley Delta-function (pp. 4--6); the counting function of tau^+(n) and the estimate (16) for H(x, y, 2y) (pp. 7--8); the theorem on the number h_alpha(n) of k with |log log p_k(n) - k| <= alpha_k (p. 13); the densities lambda_k(p) and Lambda_k(d) of integers whose k-th prime factor or k-th divisor is given (pp. 9--13); the density of integers with one divisor in (y, z], and the gaps between integers with a divisor in (n, 2n] (pp. 17--18); the largest prime factors of n and n + 1 (pp. 18--19); and the number of totient values and prime chains (p. 19).

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