On the number of integers in a generalized multiplication table

TL;DR

This paper extends the understanding of the number of distinct products in generalized multiplication tables for multiple variables, providing order of magnitude results for various parameter ranges and discussing limitations of current methods.

Contribution

It generalizes previous results by establishing order of magnitude for the count of products for arbitrary parameters when k=2,3,4,5, and offers partial results for larger k, along with heuristic insights.

Findings

Determined order of magnitude for k=2,3,4,5 cases.

Provided partial results for k>5.

Developed heuristic explaining method limitations.

Abstract

Motivated by the Erdos multiplication table problem we study the following question: Given numbers N_1,...,N_{k+1}, how many distinct products of the form n_1...n_{k+1} with n_i<N_i for all i are there? Call A_{k+1}(N_1,...,N_{k+1}) the quantity in question. Ford established the order of magnitude of A_2(N_1,N_2) and the author of A_{k+1}(N,...,N) for all k>1. In the present paper we generalize these results by establishing the order of magnitude of A_{k+1}(N_1,...,N_{k+1}) for arbitrary choices of N_1,...,N_{k+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.