Roots of unity and nullity modulo
Authors:
Steven Finch, Greg Martin and Pascal Sebah
Journal:
Proc. Amer. Math. Soc. 138 (2010), 27292743
MSC (2010):
Primary 11N37; Secondary 11M45
Published electronically:
March 25, 2010
MathSciNet review:
2644888
Fulltext PDF Free Access
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Additional Information
Abstract: For a fixed positive integer , we consider the function of that counts the number of elements of order in . We show that the average growth rate of this function is for an explicitly given constant , where is the number of divisors of . From this we conclude that the average growth rate of the number of primitive Dirichlet characters modulo of order is for . We also consider the number of elements of whose th power equals 0, showing that its average growth rate is for another explicit constant . Two techniques for evaluating sums of multiplicative functions, the WirsingOdoni and SelbergDelange methods, are illustrated by the proofs of these results.
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 4.
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 5.
 G. Greaves, Sieves in Number Theory, SpringerVerlag, Berlin, 2001. MR 1836967 (2002i:11092)
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 F. Luca and I. E. Shparlinski, Average multiplicative orders of elements modulo , Acta Arith. 109 (2003) 387411. MR 2009051 (2004i:11113)
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 H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory. I. Classical Theory, Cambridge University Press, Cambridge, 2007. MR 2378655 (2009b:11001)
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 P. Moree, On the average number of elements in a finite field with order or index in a prescribed residue class, Finite Fields Appl. 10 (2004) 438463. MR 2067608 (2005f:11219)
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Additional Information
Steven Finch
Affiliation:
Department of Statistics, Harvard University, Cambridge, Massachusetts 021382901
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/S0002993910103414
Received by editor(s):
August 31, 2009
Received by editor(s) in revised form:
December 11, 2009
Published electronically:
March 25, 2010
Communicated by:
WenChing Winnie Li
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
