On the Number of Integers in a Generalized Multiplication Table

Motivated by the Erdős multiplication table problem we study the following question: Given numbers N1, . . . , Nk+1, how many distinct products of the form n1 · · ·nk+1 with 1 ≤ ni ≤ Ni for i ∈ {1, . . . , k + 1} are there? Call Ak+1(N1, . . . , Nk+1) the quantity in question. Ford established the order of magnitude of A2(N1, N2) and the author of Ak+1(N, . . . , N) for all k ≥ 2. In the present paper we generalize these results by establishing the order of magnitude of Ak+1(N1, . . . , Nk+1) for arbitrary choices of N1, . . . , Nk+1 when 2 ≤ k ≤ 5. Moreover, we obtain a partial answer to our question when k ≥ 6. Lastly, we develop a heuristic argument which explains why the limitation of our method is k = 5 in general and we suggest ways of improving the results of this paper.