On the distribution of the order and index of g(mod p) over residue classes III

Details On the distribution of the order and index of g(mod p) over residue classes III On the distribution of the order and index of g(mod p) over residue classes III

2004-05-27

arxiv.org/abs/math/0405527v1

J. Number Theory 120 (2006), 132-160

Mathematics

Number Theory

21 pages

Scientific paper

10.1016/j.jnt.2005.11.005

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.

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