Roots of unity and nullity modulo

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 | References | Similar Articles | 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 Wirsing-Odoni and Selberg-Delange methods, are illustrated by the proofs of these results.

**1.**Gautami Bhowmik and Jan-Christoph Schlage-Puchta,*Natural boundaries of Dirichlet series*. part 1, Funct. Approx. Comment. Math.**37**(2007), no. part 1, 17–29. MR**2357306**, 10.7169/facm/1229618738**2.**Chantal David, Jack Fearnley, and Hershy Kisilevsky,*On the vanishing of twisted 𝐿-functions of elliptic curves*, Experiment. Math.**13**(2004), no. 2, 185–198. MR**2068892****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.**George Greaves,*Sieves in number theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 43, Springer-Verlag, Berlin, 2001. MR**1836967****6.**B. V. Levin and A. S. Faĭnleĭb,*Application of certain integral equations to questions of the theory of numbers*, Uspehi Mat. Nauk**22**(1967), no. 3 (135), 119–197 (Russian). MR**0229600****7.**Florian Luca and Igor E. Shparlinski,*Average multiplicative orders of elements modulo 𝑛*, Acta Arith.**109**(2003), no. 4, 387–411. MR**2009051**, 10.4064/aa109-4-7**8.**Hugh L. Montgomery and Robert C. Vaughan,*Multiplicative number theory. I. Classical theory*, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR**2378655****9.**Pieter Moree,*Approximation of singular series and automata*, Manuscripta Math.**101**(2000), no. 3, 385–399. With an appendix by Gerhard Niklasch. MR**1751040**, 10.1007/s002290050222**10.**Pieter 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), no. 3, 438–463. MR**2067608**, 10.1016/j.ffa.2003.10.001**11.**P. Moree, Values of the Euler phi function not divisible by a prescribed odd prime, http://arxiv.org/abs/math/0611509.**12.**Pieter Moree and Jilyana Cazaran,*On a claim of Ramanujan in his first letter to Hardy*, Exposition. Math.**17**(1999), no. 4, 289–311. MR**1734249****13.**Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery,*An introduction to the theory of numbers*, 5th ed., John Wiley & Sons, Inc., New York, 1991. MR**1083765****14.**R. W. K. Odoni,*A problem of Rankin on sums of powers of cusp-form coefficients*, J. London Math. Soc. (2)**44**(1991), no. 2, 203–217. MR**1136435**, 10.1112/jlms/s2-44.2.203**15.**R. W. K. Odoni,*Solution of a generalised version of a problem of Rankin on sums of powers of cusp-form coefficients*, Acta Arith.**104**(2002), no. 3, 201–223. MR**1914720**, 10.4064/aa104-3-1**16.**J. H. Rickert,*Solutions Manual to Accompany NZM 5th ed.*, unpublished manuscript (available from H. L. Montgomery).**17.**M. du Sautoy, Zeta functions of groups and natural boundaries, unpublished manuscript (2000), available at http://people.maths.ox.ac.uk/~dusautoy/1hard/prepri.htm.**18.**Daniel Shanks,*Solved and unsolved problems in number theory*, 2nd ed., Chelsea Publishing Co., New York, 1978. MR**516658****19.**Blair K. Spearman and Kenneth S. Williams,*Values of the Euler phi function not divisible by a given odd prime*, Ark. Mat.**44**(2006), no. 1, 166–181. MR**2237219**, 10.1007/s11512-005-0001-6**20.**Gérald Tenenbaum,*Introduction to analytic and probabilistic number theory*, Cambridge Studies in Advanced Mathematics, vol. 46, Cambridge University Press, Cambridge, 1995. Translated from the second French edition (1995) by C. B. Thomas. MR**1342300**

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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:
https://doi.org/10.1090/S0002-9939-10-10341-4

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.