A short-interval Hildebrand-Tenenbaum theorem

In the late eighties, Hildebrand and Tenenbaum proved an asymptotic formula for the number of positive integers below x, having exactly ν distinct prime divisors: πν(x) ~ x δν(x). Here we consider the restricted count πν(x,y) for integers lying in the short interval $[x,x+y]$. Extending the result of Hildebrand-Tenenbaum to this setting, we show that for any ε >0 \[ πν(x,y) ~ y δν(x), \] uniformly over all 1 ≤ ν ≤ log x/(log log x)4+ε and all x7/12 +ε ≤ y ≤ x. Consequently, we obtain mean upper bounds for the k-fold divisor function τₖ in short intervals, with strong uniformity in k.