Resolution of Erdős' problems about unimodularity

Abstract

Letting ${\delta }_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$, Erdős 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 $p$.

arXiv comment5 pages