Character sums for primitive root densities

Peter Stevenhagen

Description

Character sums for primitive root densities Peter Stevenhagen Geocrypt, Bastia June 28, 2011

• proved around


• field


• function field analogues


• primitive root modulo


• infinitely many primes


• artin's


• bilharz proved

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 32
• Langue: English

Extrait

Character

sums for primitive

Peter

Stevenhagen

Geocrypt, June 28,

Bastia 2011

root

densities

Artin’s primitive root conjecture

We calla∈Zaprimitive rootmodulo a primepif we have

F p ∗ =hamodpi.

Supposea∈Zis an integer that is not an exact power.

Artin’s conjecture (1927): The set of primes p for which a is a primitive root modulo p is infinite, with natural density

Y

qprime

1 (1−)≈.3739558. q(q−1)

The index

Heuristic derivation

[F ∗ p :hamodpi]

is divisible by a primeqif and only ifpsplits completely in the number field

K q =Q(ζ q , √ q a) =Split Q (X q −a),

of degree[K q :Q] =q(q−1). Forfixedq, a fraction q(q 1 −1) of all primespis eliminated.

Now“take the limit”over all primesq.