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
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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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 G. Bhowmik and J.C. SchlagePuchta, Natural boundaries of Dirichlet series, Funct. Approx. Comment. Math. 37 (2007) 1729. MR 2357306 (2008j:11116)
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 C. David, J. Fearnley and H. Kisilevsky, On the vanishing of twisted functions of elliptic curves, Experiment. Math. 13 (2004) 185198. MR 2068892 (2005e:11082)
 3.
 S. Finch, Quartic and octic characters modulo , http://arxiv.org/abs/0907.4894.
 4.
 S. Finch and P. Sebah, Squares and cubes modulo , http://arxiv.org/abs/math/0604465.
 5.
 G. Greaves, Sieves in Number Theory, SpringerVerlag, Berlin, 2001. MR 1836967 (2002i:11092)
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 B. V. Levin and A. S. Fainleib, Application of certain integral equations to questions of the theory of numbers (Russian), Uspehi Mat. Nauk 22 (1967) n. 3, 119197. Engl. transl. in Russian Math. Survey 22 (1967) n. 3, 119204. MR 0229600 (37:5174)
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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)
 8.
 H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory. I. Classical Theory, Cambridge University Press, Cambridge, 2007. MR 2378655 (2009b:11001)
 9.
 P. Moree, Approximation of singular series and automata, with an appendix by Gerhard Niklasch, Manuscripta Math. 101 (2000) 385399. MR 1751040 (2001f:11204)
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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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 P. Moree, Values of the Euler phi function not divisible by a prescribed odd prime, http://arxiv.org/abs/math/0611509.
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 P. Moree and J. Cazaran, On a claim of Ramanujan in his first letter to Hardy, Exposition. Math. 17 (1999) 289311. MR 1734249 (2001c:11103)
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 I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers, 5th ed., John Wiley & Sons, Inc., New York, 1991. MR 1083765 (91i:11001)
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 R. W. K. Odoni, Solution of a generalised version of a problem of Rankin on sums of powers of cuspform coefficients, Acta Arith. 104 (2002) 201223. MR 1914720 (2003m:11067)
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 D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd ed., Chelsea Publishing Co., New York, 1978. MR 0516658 (80e:10003)
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 G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, Cambridge, 1995. MR 1342300 (97e:11005b)
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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
PII:
S 00029939(10)103414
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.
