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On the number of integers in a generalized multiplication table

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Veröffentlicht/Copyright: 12. Juli 2012
Journal für die reine und angewandte Mathematik (Crelles Journal)
Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2014 Heft 689

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

© 2014 by Walter de Gruyter Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Nonlinear PDE aspects of the tt* equations of Cecotti and Vafa
  3. On the number of integers in a generalized multiplication table
  4. Peripheral structures of relatively hyperbolic groups
  5. What is the total Betti number of a random real hypersurface?
  6. Parametrization of ideal classes in rings associated to binary forms
  7. Weakly-exceptional singularities in higher dimensions
  8. Erratum to A derived approach to geometric McKay correspondence in dimension three (J. reine angew. Math. 636 (2009), 193–236)
https://doi.org/10.1515/crelle-2012-0064

Artikel in diesem Heft

  1. Frontmatter
  2. Nonlinear PDE aspects of the tt* equations of Cecotti and Vafa
  3. On the number of integers in a generalized multiplication table
  4. Peripheral structures of relatively hyperbolic groups
  5. What is the total Betti number of a random real hypersurface?
  6. Parametrization of ideal classes in rings associated to binary forms
  7. Weakly-exceptional singularities in higher dimensions
  8. Erratum to A derived approach to geometric McKay correspondence in dimension three (J. reine angew. Math. 636 (2009), 193–236)
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