A short-interval Hildebrand-Tenenbaum theorem

TL;DR

This paper extends the Hildebrand-Tenenbaum theorem to short intervals, providing uniform asymptotic formulas for counting integers with a fixed number of prime divisors and bounds for divisor functions.

Contribution

It establishes a short-interval version of the Hildebrand-Tenenbaum theorem with uniform asymptotics and bounds for divisor functions, advancing understanding of prime divisor distributions.

Findings

Asymptotic equivalence for $ u$-prime divisor counts in short intervals

Uniform results valid for a wide range of $ u$ and interval lengths

Mean upper bounds for the divisor function $ au_k$ in short intervals

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 $\varepsilon >0$, the asymptotic equivalence \[ \pi_{\nu}(x,y) \sim y \delta_{\nu}(x)\] holds uniformly over all $1 \le \nu \le (\log x)^{1/3}/(\log \log x)^2$ and all $x^{17/30 + \varepsilon} \leq y \leq x$. The methods also furnish mean upper bounds for the $k$-fold divisor function $\tau_k$ in short intervals, with strong uniformity in $k$.