On the density of some sets of primes III

On the distribution of the order and index of g(modp) over residue classes—I

“…As already pointed out in Section 2.5.1 these cases have been well studied. The best known error terms are due to Wiertelak [33]. The densities for these cases can be explicitly evaluated using Lemma 1.…”

“…In this direction we especially like to mention Wiertelak, who wrote many papers on this subject, starting in the 1970s of the previous century. See [33] for his most recent paper. Again one can prove that the density of the set of such primes exists and is rational.…”

Improvement of an estimate of H. Müller involving the order of 2(mod u) II

“…It turns out that the set P g (d) of primes p ∈ S(g) such that d|ord g (p) has a natural density, which will be denoted by δ g (d). This density was first determined by Wiertelak [9], who derived a rather complicated 132 P. Moree arch. math.…”

“…The following result is due to Wiertelak [9] (the exponent ω(d) + 3 in the log log x term given here is unfortunately misquoted in Moree [4] as ω(d) + 1). As usual the logarithmic integral is denoted by Li(x).…”

“…R e m a r k. Wiertelak [9] studied P (x; g, d) in depth in case d is squarefree. Note (as did Wiertelak) that if d is squarefree, then P (x; g, d) counts the number of primes p x with ν p (g) = 0 such that the congruence g dy ≡ g(mod p) has a solution y.…”