On the number of integers in a generalized multiplication table

Dimitris Koukoulopoulos

doi:10.1515/crelle-2012-0064

Published by De Gruyter July 12, 2012

On the number of integers in a generalized multiplication table

Dimitris Koukoulopoulos

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2012-0064

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Abstract.

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 the one 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.

Funding source: NSF

Award Identifier / Grant number: DMS 05-55367, DMS 08-38434, DMS 09-01339

I would like to thank Kevin Ford for many valuable suggestions as well as for discussions that led to an earlier version of Lemma 2.3.

Received: 2011-2-16

Revised: 2012-5-15

Published Online: 2012-7-12

Published in Print: 2014-4-1

On the number of integers in a generalized multiplication table

Abstract.

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