Abstract
We extend to an arbitrary number field the best known bounds towards Ramanujan for the group $\mathrm{GL}_n$, $n=2,3,4$. In particular, we present a technique which overcomes the analytic obstacles posed by the presence of an infinite group of units.
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TITLE = {Refined estimates towards the {R}amanujan and {S}elberg conjectures},
NOTE = {appendix to H. Kim, Functoriality for the exterior square of {${\rm GL}\sb 4$} and the symmetric fourth of {${\rm GL}\sb 2$}},
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}