Resolution of Erdős' problems about unimodularity

2025/01/17 by Cambie, Stijn
#11A41 #11A51 #11K36 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2501.10333

Abstract

Letting δ₁(n,m) be the density of the set of integers with exactly one divisor in (n,m), Erdős wondered if δ₁(n,m) is unimodular for fixed n. We prove this is false in general, as the sequence (δ₁(n,m)) has superpolynomially many local extrema. However, we confirm unimodality in the single case for which it occurs; n = 1. We also solve the question on unimodality of the density of integers whose k^th prime is p.

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