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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Roots of unity and nullity modulo $ n$


Authors: Steven Finch, Greg Martin and Pascal Sebah
Journal: Proc. Amer. Math. Soc. 138 (2010), 2729-2743
MSC (2010): Primary 11N37; Secondary 11M45
Published electronically: March 25, 2010
MathSciNet review: 2644888
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Abstract: For a fixed positive integer $ \ell$, we consider the function of $ n$ that counts the number of elements of order $ \ell$ in $ \mathbb{Z}_n^*$. We show that the average growth rate of this function is $ C_\ell(\log n)^{d(\ell)-1}$ for an explicitly given constant $ C_\ell$, where $ d(\ell)$ is the number of divisors of $ \ell$. From this we conclude that the average growth rate of the number of primitive Dirichlet characters modulo $ n$ of order $ \ell$ is $ (d(\ell)-1)C_\ell (\log n)^{d(\ell)-2}$ for $ \ell\ge2$. We also consider the number of elements of $ \mathbb{Z}_n$ whose $ \ell$th power equals 0, showing that its average growth rate is $ D_\ell(\log n)^{\ell-1}$ for another explicit constant $ D_\ell$. Two techniques for evaluating sums of multiplicative functions, the Wirsing-Odoni and Selberg-Delange methods, are illustrated by the proofs of these results.


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Additional Information

Steven Finch
Affiliation: Department of Statistics, Harvard University, Cambridge, Massachusetts 02138-2901
Email: Steven.Finch@inria.fr

Greg Martin
Affiliation: Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2
Email: gerg@math.ubc.ca

Pascal Sebah
Affiliation: DS Research, Dassault Systèmes, Suresnes, France
Email: PSebah@yahoo.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10341-4
PII: S 0002-9939(10)10341-4
Received by editor(s): August 31, 2009
Received by editor(s) in revised form: December 11, 2009
Published electronically: March 25, 2010
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.