A short-interval Hildebrand-Tenenbaum theorem

Abstract

In the late eighties, Hildebrand and Tenenbaum proved an asymptotic formula for the number of positive integers below $x$, having exactly $\nu$ distinct prime divisors: ${\pi }_{\nu }(x)\sim x{\delta }_{\nu }(x)$. Here we consider the restricted count ${\pi }_{\nu }(x,y)$ for integers lying in the short interval $(x,x+y]$. In this setting, we show that for any $\epsilon >0$, the asymptotic equivalence \[ π_ν(x,y) \sim y δ_ν(x)\] holds uniformly over all $1\le \nu \le (\log x{)}^{1/3}/(\log \log x{)}^2$ and all $x^{17/30+\epsilon }\le y\le x$. The methods also furnish mean upper bounds for the $k$-fold divisor function ${\tau }_k$ in short intervals, with strong uniformity in $k$.

Updated, following referee's suggestions and comments