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ON THE LEAST SQUAREFREE NUMBER IN AN ARITHMETIC PROGRESSION

Part of: Elementary number theory Multiplicative number theory

Published online by Cambridge University Press:  23 March 2017

R. M. Nunes
R. M. Nunes*
Affiliation:
EPFL SB MATHGEOM TAN, Station 8, CH-1015 Lausanne, Switzerland email ramon.moreiranunes@epfl.ch
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Abstract

We prove an asymptotic formula for squarefree numbers in arithmetic progressions, improving previous results by Prachar and Hooley. As a consequence we improve a lower bound of Heath-Brown for the least squarefree number in an arithmetic progression.

MSC classification

Primary: 11N37: Asymptotic results on arithmetic functions
Secondary: 11A07: Congruences; primitive roots; residue systems
Type
Research Article
Information
Mathematika , Volume 63 , Issue 2 , 2017 , pp. 483 - 498
Copyright
Copyright © University College London 2017 

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References

Erdős, P., Über die kleinste quadratfreie Zahl einer arithmetischen Reihe. Monatsh. Math. 64 1960, 314–316 (in German).CrossRefGoogle Scholar
Heath-Brown, D. R., The least square-free number in an arithmetic progression. J. Reine Angew. Math. 332 1982, 204–220.Google Scholar
Hooley, C., A note on square-free numbers in arithmetic progressions. Bull. Lond. Math. Soc. 7 1975, 133–138.CrossRefGoogle Scholar
Nunes, R. M., Squarefree numbers in large arithmetic progressions. Preprint, 2016,arXiv:1602.00311.CrossRefGoogle Scholar
Pierce, L. B., The 3-part of class numbers of quadratic fields. J. Lond. Math. Soc. (2) 71(3) 2005, 579–598.CrossRefGoogle Scholar
Prachar, K., Über die kleinste quadratfreie Zahl einer arithmetischen Reihe. Monatsh. Math. 62 1958, 173–176 (in German).CrossRefGoogle Scholar