Let$\mathbb{K}$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$\sum_{\mathcal{N}\mathfrak{p} \leq x} {\mathcal{I}_{\gamma}^\kappa({\mathfrak{p})$.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.