We consider a downlink multi-user scenario and investigate the use of reconfigurable intelligent surfaces (RISs) to maximize the dirty-paper-coding (DPC) sum rate of the RIS-assisted broadcast channel. Different from prior works, which maximize the rate achievable by linear precoders, we assume a capacity-achieving DPC scheme is employed at the transmitter and optimize the transmit covariances and RIS reflection coefficients to directly maximize the sum capacity of the broadcast channel. We propose an optimization algorithm that iteratively alternates between optimizing the transmit covariances using convex optimization and the RIS reflection coefficients using Riemannian manifold optimization. Our results show that the proposed technique can be used to effectively improve the sum capacity in a variety of scenarios compared to benchmark schemes.