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Artin's primitive root conjecture for quadratic fields

Hans Roskam (2002)

Journal de théorie des nombres de Bordeaux

Fix an element α in a quadratic field K . Define S as the set of rational primes p , for which α has maximal order modulo p . Under the assumption of the generalized Riemann hypothesis, we show that S has a density. Moreover, we give necessary and sufficient conditions for the density of S to be positive.

Chebotarev sets

Hershy Kisilevsky, Michael O. Rubinstein (2015)

Acta Arithmetica

We consider the problem of determining whether a set of primes, or, more generally, prime ideals in a number field, can be realized as a finite union of residue classes, or of Frobenius conjugacy classes. We give necessary conditions for a set to be realized in this manner, and show that the subset of primes consisting of every other prime cannot be expressed in this way, even if we allow a finite number of exceptions.

Chebyshev's method for number fields

José Felipe Voloch (2000)

Journal de théorie des nombres de Bordeaux

We give an elementary proof of an explicit estimate for the number of primes splitting completely in an extension of the rationals. The proof uses binomial coefficents and extends Chebyshev's classical approach.

On ideals free of large prime factors

Eira J. Scourfield (2004)

Journal de Théorie des Nombres de Bordeaux

In 1989, E. Saias established an asymptotic formula for Ψ ( x , y ) = n x : p n p y with a very good error term, valid for exp ( log log x ) ( 5 / 3 ) + ϵ y x , x x 0 ( ϵ ) , ϵ > 0 . We extend this result to an algebraic number field K by obtaining an asymptotic formula for the analogous function Ψ K ( x , y ) with the same error term and valid in the same region. Our main objective is to compare the formulae for Ψ ( x , y ) and Ψ K ( x , y ) , and in particular to compare the second term in the two expansions.

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