primes with primitive root 10

[Pieter Moree]

Klaus Brockhaus wrote:

>
> are there any results concerning the density of primes with primitive
root
> 10 (A001913), or of Long period primes, (A006883) relative to the
sequence
> of all primes (A000040)?
>
> and the same question for primes with primitive root 2 (A001122)?
Sure there are.

Assuming the Generalized Riemann Hypothesis it can be shown that
the density of primes p such that a prescribed integer g has
order (p-1)/t, with t fixed exists and, moreover, it can be
computed. This density will be a rational number times the
so called Artin constant.
  For 2 and 10 the density of primitive roots is A, the Artin
constant itself.

For a survey, vide
http://turing.wins.uva.nl/~moree/varia.htm
(Artin's primitive root conjecture - a survey)

Related questions are do the primes for which the order is
even or odd have a density ?
For example, what is the density of primes p such that
1/p has an even decimal period.
 Here typically the density is a rational number and this
can be proved unconditionally.

The long period primes, are the primes p such that 1/p has period
p-1, a litte thought reveals these are the primes p such that
10 is a primitive root modulo p.

A related problem I devoted some time on was the following:

Let g be some fixed integer and d a modulus.
As p varies over the primes, how is the order of g(modulo p)
distributed over the various congruence classes modulo d ?
I studied this in three papers (to be found on the ArXiv):
On the distribution of the order and index of g(mod p) over
residue classes --I, II, III
(of which the first will shortly appear in J. Number Theory)