Resolution of Erd\H{o}s' problems about unimodularity

Stijn Cambie

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arXiv · 2501.10333

Resolution of Erd\H{o}s' problems about unimodularity

Stijn Cambie

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Abstract

Letting $\delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$, Erd\H{o}s wondered if $\delta_1(n,m)$ is unimodular for fixed $n$. We prove this is false in general, as the sequence $(\delta_1(n,m))$ 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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Stijn Cambie. 2025-01-17. Resolution of Erd\H{o}s' problems about unimodularity. https://arxiv.org/abs/2501.10333

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