Given a fixed integer $k$ and a large positive integer $N$, I'd like to count the number of positive integer solutions to the equation $\prod_{i=1}^{k}x_i\le N$ in sublinear time.
Note that when $k=2$, this is the classic Dirichlet's divisor problem which can be solved in $O(N^\frac{1}{3})$ using an algorithm described by the Polymath project here or using the convex hull technique. But it seems hard to extend such method to higher dimensions keeping asymptotically the same complexity (or within logarithms); for example, for $k=3$ the naive extension yields $O(N^\frac{5}{9})$ complexity and as $k$ goes infinity, the time complexity approaches $O(N^{1-\epsilon})$.