Generalized Erdős multiplication table problem

Question

Consider multiplication operation $$f(x_1,\dots, x_k)=\prod_{i=1}^kx_i$$ where $x_i\in\{1,\dots, n_i\}$ with $n_1,\dots, n_k\in\{1,\dots,\infty\}$.

What is the cardinality of the range?

At $k =2$ with $n_1=n_2$ this is the standard Erdos multiplication table problem whose estimates are in Distinct numbers in multiplication table.

this should be easy to look up. i know it's been done before — mathworker21, May 06 '20 at 01:05

score 4 · Accepted Answer · answered May 06 '20 at 05:51

4

This problem is solved for $k\leq 5$ and open for $k\geq 6$. See Koukoulopoulos's paper for more details.

answered May 06 '20 at 05:51

GH from MO

1 Answers1