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

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

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.

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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.