bibkey: tenenbaum2013unconventional authors: Gérald Tenenbaum year: 2013 title: “Some of Erdős’ Unconventional Problems in Number Theory, Thirty-four Years Later” doi: 10.1007/978-3-642-39286-3_23 claim: “Erdős (1979), restated as open by Tenenbaum (2013): Perhaps ε₁(n,m) is unimodular for m > n+1, but I know nothing about this.” strata_touched:

• D5/S3/Arith/Density/ErdosExactlyOneDivisorUnimodalityRefutation license: citation-only triage: anchor

Unimodality of the one-divisor interval density

Tenenbaum revisits Erdős’s questions about the set of multiples of an interval. He reproduces the 1979 suggestion and later states that the unimodality question remained open to his knowledge.

Verified locator

• DOI: https://doi.org/10.1007/978-3-642-39286-3_23

• Archived chapter PDF: https://web.archive.org/web/20220710081934id_/https://tenenb.perso.math.cnrs.fr/PPP/Erdos-100.pdf

• The sentence occurs on the chapter’s printed page 18 (volume page 668), in the discussion of the set of multiples of an interval begun on printed page 17. It reads verbatim:

  To my knowledge, the question of unimodality of ε₁(y,z) as a function of z is still open.

• The original is P. Erdős, Some unconventional problems in number theory, Astérisque 61 (1979), printed page 78: https://users.renyi.hu/~p_erdos/1979-21.pdf

  Perhaps ε₁(n,m) is unimodular for m > n+1, but I know nothing about this.

The chapter uses the interval convention y < d <= z, while the quoted 1979 sentence uses n < d < m. The Lean statement follows the original strict upper endpoint and addresses only the unqualified printed suggestion.