On the number of integers in a generalized multiplication table
arXiv:1102.3236 · doi:10.1515/crelle-2012-0064
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.
arXiv comment65 pages. Minor changes. To appear at J. Reine Angew. Math. The final publication is available at www.reference-global.com