To start with, the proofs of the theorems are long and so not suitable for explaining here in a short survey. The same holds for the display of the contents/statements of the theorems. In order to get an impression what it is all about, we provide the introduction of the article, as follows.
Introduction: “Consider a number field \(K\) and finitely many algebraic numbers \(\alpha_1,\dots,\alpha_r\in K^\times\) which generate a multiplicative subgroup of \(K^\times\) of positive rank \(r\). Let \(\mathfrak{p}\) be a prime of \(K\) such that for each \(i\) the reduction of \(\alpha_i\) modulo \(\mathfrak{p}\) is a well-defined element of \(k^\times_{\mathfrak{p}}\) (where \(k_{\mathfrak{p}}\) is the residue field at \(\mathfrak{p}\)). We study the set of primes such that for each \(i\) the multiplicative order of \((\alpha_i\bmod\mathfrak{p})\) lies in a given arithmetic progression.
More precisely, write \(\text{ord}_{\mathfrak{p}}(\alpha_i)\) for the order of \((\alpha_1\bmod\mathfrak{p})\). We will prove under GRH the existence of the density of primes \(\mathfrak{p}\) satisfying \(\text{ord}_{\mathfrak{p}}(\alpha_i)\equiv a_i\bmod d_i\) for each \(i\), where \(a_i\), \(d_i\) are some fixed integers. In Theorem 1 we give an asymptotic formula for the number of such primes. We also study the density of primes satisfying conditions on the index. Write \(\text{ind}_{\mathfrak{p}}(\alpha_i)\) for the index of the subgroup generated by \((\alpha_i\bmod\mathfrak{p})\) in \(k^\times_{\mathfrak{p}}\). Notice that \(\text{ind}_{\mathfrak{p}}(\alpha_i)= (N\mathfrak{p}- 1)/\text{ord}_{\mathfrak{p}}(\alpha_i)\). We prove the existence of the density of primes \(\mathfrak{p}\) such that \(\text{ind}_{\mathfrak{p}}(\alpha_i)= t_i\) for each \(i\), where the \(t_i\)’s are positive integers, and more generally such that \(\text{ind}_{\mathfrak{p}}(\alpha_i)\) lies in a given sequence of integers. Given a finite Galois extension \(K\), a condition on the conjugacy class of Frobenius automorphisms of the primes lying above \(\mathfrak{p}\) may also be introduced.
These results are generalizations of Ziegler’s work [V. Ziegler, Unif. Distrib. Theory 1, No. 1, 65–85 (2006; Zbl 1147.11054)] from 2006, which concerns the case of rank 1. Moreover, in [A. Perucca and P. Sgobba, Int. J. Number Theory 15, No. 8, 1617–1633 (2019; Zbl 1451.11123)], the author and Perucca have generalized Ziegler’s results to study the set of primes for which the order of the reduction of a finitely generated group of algebraic numbers lies in a given arithmetic progression, and in [A. Perucca and P. Sgobba, Unif. Distrib. Theory 15, No. 1, 75–92 (2020; Zbl 1475.11211)], they have investigated properties of the density of this set. Notice that problems of this kind have been studied in various papers by K. Chinen and L. Murata [Dev. Math. 15, 11–22 (2006; Zbl 1200.11076)], and by P. Moree [J. Number Theory 120, No. 1, 132–160 (2006; Zbl 1203.11066)], and that they are related to Artin’s Conjecture on primitive roots, see the survey [Integers 12, No. 6, 1305–1416, A13 (2012; Zbl 1271.11002)] by P. Moree.”