A short-interval Hildebrand-Tenenbaum theorem

Paper

Abstract

In the late eighties, Hildebrand and Tenenbaum proved an asymptotic formula for the number of positive integers below

, having exactly

distinct prime divisors:

. Here we consider the restricted count

for integers lying in the short interval

. In this setting, we show that for any

, the asymptotic equivalence

holds uniformly over all

and all

. The methods also furnish mean upper bounds for the

-fold divisor function

in short intervals, with strong uniformity in