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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: shuguang@hawaii.edu
Carl Pomerance
Affiliation:
Carl Pomerance, Mathematics Department, Dartmouth College, Hanover, NH 03755-3551, U.S.A., E-mail: carl.pomerance@dartmouth.edu
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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

Information

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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References

1.Carmichael, R. D., The Theory of Numbers, Wiley (New York, 1914).Google Scholar
2.Hooley, C., On Artin's conjecture. J. Reine Angew. Math. 225 (1967), 209–220.Google Scholar
3.Iwaniec, H. and Kowalski, E., Analytic Number Theory, American Mathematical Society (Providence, RI, 2004).Google Scholar
4.Li, S., On extending Artin's conjecture to composite moduli. Mathematika 46 (1999), 373–390.CrossRefGoogle Scholar
5.Li, S., Artin's conjecture on average for composite moduli. J. Number Theory 84 (2000), 93–118.CrossRefGoogle Scholar
6.Li, S., An improvement of Artin's conjecture on average for composite moduli. Mathematika 51 (2004), 97–109.CrossRefGoogle Scholar
7.Li, S. and Pomerance, C., Primitive roots: a survey. In Number Theoretic Methods-Future Trends (Developments Mathematics 8) (eds. Jia, C. and Kanemitsu, S.), Kluwer Academic (Dordrecht, 2002), 219–231.CrossRefGoogle Scholar
8.Li, S. and Pomerance, C., On generalizing Artin's conjecture on primitive roots to composite moduli. J. Reine Angew. Math. 556 (2003), 205–.Google Scholar
9.Luca, F. and Pomerance, C., On the average number of divisors of the Euler function. Publ. Math. Debrecen 70 (2007), 125–148.CrossRefGoogle Scholar
10.Martin, G., The least prime primitive root and the shifted sieve. Acta Arith. 80 (1997), 277–288.CrossRefGoogle Scholar
11.Moree, P., Artin's primitive root conjecture — a survey. Preprint, 2004, math.NT/0412262.Google Scholar
12.Murty, M. R., Artin's conjecture for primitive roots. Math. Intelligencer 10 (1988), 59–67.CrossRefGoogle Scholar
13.Stephens, P. J., An average result for Artin's conjecture. Mathematika 16 (1969), 178–188.CrossRefGoogle Scholar