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Character sums for primitive root densities

De
33 pages
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


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Character
sums for primitive
Peter
Stevenhagen
Geocrypt, June 28,
Bastia 2011
root
densities
Artin’s primitive root conjecture
We callaZaprimitive rootmodulo a primepif we have
Fp=hamodpi.
SupposeaZis 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(q1)
The index
Heuristic derivation
[Fp:hamodpi]
is divisible by a primeqif and only ifpsplits completely in the number field
Kq=Q(ζq,qa) =SplitQ(Xqa),
of degree[Kq:Q] =q(q1). Forfixedq, a fractionq(q11)of all primespis eliminated.
Now“take the limit”over all primesq.