On the distribution of the order and index for the reductions of algebraic numbers

Let α₁,…,α_r be algebraic numbers in a number field K generating a subgroup of rank r in K^×. We investigate under GRH the number of primes p of K such that each of the orders of (α_imodp) lies in a given arithmetic progression associated to α_i. We also study the primes p for which the index of (α_imodp) is a fixed integer or lies in a given set of integers for each i. An additional condition on the Frobenius conjugacy class of p may be considered. Such results are generalizations of a theorem of Ziegler from 2006, which concerns the case r=1 of this problem.