Artin's primitive root conjecture and a problem of Rohrlich

Abstract

AbstractLet$\mathbb{K}$be a number field, Γ a finitely generated subgroup of$\mathbb{K}$*, for instance the unit group of$\mathbb{K}$, and κ>0. For an ideal$\mathfrak{a}$of$\mathbb{K}$let indΓ($\mathfrak{a}$]></alt-text></inline-graphic>) denote the multiplicative index of the reduction of Γ in <inline-graphic name="S0305004114000206_inline3"><alt-text><![CDATA[$(\mathcal{O}_\mathbb{K}/\mathfrak{a})$* (whenever it makes sense). For a prime ideal$\mathfrak{p}$of$\mathbb{K}$and a positive integer γ let$\mathcal{I}_\gamma^\kappa(\mathfrak{p})$be the average of${ind}_{\langle a_1,\dots,a_\gamma\rangle}(\mathfrak{p})^\kappa$over all tupels$(a_1,\dots,a_\gamma)\in{(\mathcal{O}_\mathbb{K}/\mathfrak{p})^*}^\gamma$. Motivated by a problem of Rohrlich we prove, partly conditionally on fairly standard hypotheses, lower bounds for$\sum_{\mathcal{N}{\mathfrak{a}\leq x}{ind}_{\Gamma}({\mathfrak{a})^\kappa$and asymptotic formulae for$\sum_{\mathcal{N}\mathfrak{p} \leq x} {\mathcal{I}_{\gamma}^\kappa({\mathfrak{p})$.