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The Artin–Carmichael Primitive Root Problem on Average
Author(s) -
Li Shuguang,
Pomerance Carl
Publication year - 2009
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300000991
Subject(s) - mathematics , combinatorics , coprime integers , order (exchange) , integer (computer science) , natural number , primitive root modulo n , prime number , number theory , asymptotic formula , prime factor , prime number theorem , prime (order theory) , discrete mathematics , finance , computer science , economics , programming language
For a natural number n , let λ( n ) denote the order of the largest cyclic subgroup of (ℤ/ n ℤ)*. For a given integer a , let N a ( x ) denote the number of n ≤ x coprime to a for which a has order λ( n ) in (ℤ/ n ℤ)*. Let R ( n ) denote the number of elements of (ℤ/ n ℤ)* with order λ( n ). It is natural to compare N a ( x ) with ∑ n ≤ x R ( n )/ n . In this paper we show that the average of N a ( 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 .