  We provide accurate approximations of the sum-rate capacity of an
opportunistic time-sharing downlink, when a reconfigurable intelligent surface
(RIS) assists the transmission from a single-antenna base station (BS) to
single-antenna user equipments (UEs). We consider the fading effects of both
the direct (i.e., BS-to-UEs) and reflection (i.e, BS-to-RIS-to-UEs) links, by
developing two approximations: the former one is based on hardening of the
reflection channel for large values of the number of meta-atoms; the latter one
relies on the distribution of the sum of Nakagami variates and does not require
channel hardening. Our derivations show the dependence of the sum-rate capacity
as a function of both the number of users and the number of meta-atoms, as well
as to establish a comparison with a downlink without an RIS. Numerical results
corroborate the accuracy and validity of the mathematical analysis.
