On the number of integers in a generalized multiplication table
Abstract
Motivated by the Erdos multiplication table problem we study the following question: Given numbers N1,...,Nk+1, how many distinct products of the form n1...nk+1 with ni<Ni for all i 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>1. 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 k is 2,3,4 or 5. Moreover, we obtain a partial answer to our question when k>5. 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.
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