  For a fixed rational number g and integers a and d the sets N_g(a,d),
respectively R_g(a,d), of primes p for which the order, respectively the index
of g(mod p) is congruent to a(mod d), are considered.
  Under the Generalized Riemann Hypothesis (GRH), it was shown in part II that
these sets have a natural density. Here it is shown that these densities can be
expressed as linear combinations of certain constants introduced by Pappalardi.
  Furthermore it is proved that these densities equal their g-averages for
almost all g. It is also shown that if such a density does not equal its
g-average then it is close to it. Thus these quantities experience a strong
`pull' towards the g-average.
