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Artin's Conjecture on Average for Composite Moduli

✍ Scribed by Shuguang Li

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
201 KB
Volume
84
Category
Article
ISSN
0022-314X

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✦ Synopsis

Let a be an integer { &1 and not a square. Let P a (x) be the number of primes up to x for which a is a primitive root. Goldfeld and Stephens proved that the average value of P a (x) is about a constant multiple of xΓ‚ln x. Carmichael extended the definition of primitive roots to that of primitive *-roots for composite moduli n, which are integers with the maximal order modulo n. Let N a (x) be the number of natural numbers up to x for which a is a primitive *-root. In this paper we will prove that the average value of N a (x) oscillates. That is, lim x Γ„ 1Γ‚x 2 1 a x N a (x)>0 and x Γ„ 1Γ‚x 2 1 a x N a (x)=0.

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