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

TL;DR

This paper investigates the unimodality of certain integer density functions related to divisor counts and prime positions, disproving general unimodality and confirming it in specific cases, thus resolving longstanding questions posed by Erdős.

Contribution

It demonstrates that the density function elta_1(n,m) is not unimodal in general, but confirms unimodality when n=1, and addresses the unimodality of prime-related densities.

Findings

elta_1(n,m) has superpolynomially many local extrema.

Unimodality holds specifically for n=1.

The density of integers with the k-th prime equal to p is unimodal.

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 $p$.