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The Artin–Carmichael Primitive Root Problem on Average

Part of: Elementary number theory Multiplicative number theory

Published online by Cambridge University Press:  21 December 2009

Shuguang Li  and
Carl Pomerance
Shuguang Li
Affiliation:
Shuguang Li, Department of Mathematics, University of Hawaii at Hilo, Hilo, HI 96720-4091, U.S.A., E-mail: [email protected]
Carl Pomerance
Affiliation:
Carl Pomerance, Mathematics Department, Dartmouth College, Hanover, NH 03755-3551, U.S.A., E-mail: [email protected]
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Abstract

For a natural number n, let λ(n) denote the order of the largest cyclic subgroup of (ℤ/nℤ)*. For a given integer a, let Na(x) denote the number of n ≤ x coprime to a for which a has order λ(n) in (ℤ/nℤ)*. Let R(n) denote the number of elements of (ℤ/nℤ)* with order λ(n). It is natural to compare Na(x) with ∑n≤xR(n)/n. In this paper we show that the average of Na(x) for 1 ≤ a ≤ y is indeed asymptotic to this sum, provided y ≥ exp((2 + ε)(log x log log x)1/2), thus improving a theorem of the first author who had this for y ≥ exp((log x)3/4;). The result is to be compared with a similar theorem of Stephens who considered the case of prime numbers n.

MSC classification

Secondary: 11N37: Asymptotic results on arithmetic functions 11A07: Congruences; primitive roots; residue systems
Type
Research Article
Information
Mathematika , Volume 55 , Issue 1-2 , December 2009 , pp. 167 - 176
Copyright
Copyright © University College London 2009

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