  This paper provides a theoretical framework for understanding the performance
of reconfigurable intelligent surface (RIS)-aided massive multiple-input
multiple-output (MIMO) with zero-forcing (ZF) detectors under imperfect channel
state information (CSI). We first propose a low-overhead minimum mean square
error (MMSE) channel estimator, and then derive and analyze closed-form
expressions for the uplink achievable rate. Our analytical results demonstrate
that: $1)$ regardless of the RIS phase shift design, the rate of all users
scales at least on the order of
$\mathcal{O}\left(\log_2\left(MN\right)\right)$, where $M$ and $N$ are the
numbers of antennas and reflecting elements, respectively; $2)$ by aligning the
RIS phase shifts to one user, the rate of this user can at most scale on the
order of $\mathcal{O}\left(\log_2\left(MN^2\right)\right)$; $3)$ either $M$ or
the transmit power can be reduced inversely proportional to $N$, while
maintaining a given rate. Furthermore, we propose two low-complexity
majorization-minimization (MM)-based algorithms to optimize the sum user rate
and the minimum user rate, respectively, where closed-form solutions are
obtained in each iteration. Finally, simulation results validate all derived
analytical results. Our simulation results also show that the maximum sum rate
can be closely approached by simply aligning the RIS phase shifts to an
arbitrary user.
