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
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 WirsingOdoni and SelbergDelange methods, are illustrated by the proofs of these results.
 1.
Gautami
Bhowmik and JanChristoph
SchlagePuchta, Natural boundaries of Dirichlet series. part
1, Funct. Approx. Comment. Math. 37 (2007), no. part
1, 17–29. MR 2357306
(2008j:11116), http://dx.doi.org/10.7169/facm/1229618738
 2.
Chantal
David, Jack
Fearnley, and Hershy
Kisilevsky, On the vanishing of twisted 𝐿functions of
elliptic curves, Experiment. Math. 13 (2004),
no. 2, 185–198. 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.
George
Greaves, Sieves in number theory, Ergebnisse der Mathematik
und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)],
vol. 43, SpringerVerlag, Berlin, 2001. MR 1836967
(2002i:11092)
 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
(37 #5174)
 7.
Florian
Luca and Igor
E. Shparlinski, Average multiplicative orders of elements modulo
𝑛, Acta Arith. 109 (2003), no. 4,
387–411. MR 2009051
(2004i:11113), http://dx.doi.org/10.4064/aa10947
 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
(2009b:11001)
 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
(2001f:11204), http://dx.doi.org/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
(2005f:11219), http://dx.doi.org/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
(2001c:11103)
 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
(91i:11001)
 14.
R.
W. K. Odoni, A problem of Rankin on sums of powers of cuspform
coefficients, J. London Math. Soc. (2) 44 (1991),
no. 2, 203–217. MR 1136435
(93d:11048), http://dx.doi.org/10.1112/jlms/s244.2.203
 15.
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), no. 3, 201–223. MR 1914720
(2003m:11067), http://dx.doi.org/10.4064/aa10431
 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
(80e:10003)
 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
(2007j:11133), http://dx.doi.org/10.1007/s1151200500016
 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
(97e:11005b)
 1.
 G. Bhowmik and J.C. SchlagePuchta, Natural boundaries of Dirichlet series, Funct. Approx. Comment. Math. 37 (2007) 1729. MR 2357306 (2008j:11116)
 2.
 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)
 6.
 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)
 7.
 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)
 10.
 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)
 11.
 P. Moree, Values of the Euler phi function not divisible by a prescribed odd prime, http://arxiv.org/abs/math/0611509.
 12.
 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)
 13.
 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)
 14.
 R. W. K. Odoni, A problem of Rankin on sums of powers of cuspform coefficients, J. London Math. Soc. 44 (1991) 203217. MR 1136435 (93d:11048)
 15.
 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)
 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.
 D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd ed., Chelsea Publishing Co., New York, 1978. MR 0516658 (80e:10003)
 19.
 B. K. Spearman and K. S. Williams, Values of the Euler phi function not divisible by a given odd prime, Ark. Mat. 44 (2006) 166181. MR 2237219 (2007j:11133)
 20.
 G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, Cambridge, 1995. MR 1342300 (97e:11005b)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
11N37,
11M45
Retrieve articles in all journals
with MSC (2010):
11N37,
11M45
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.
