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Reconfigurable intelligent surface (RIS) is considered as a promising technology, which can effectively control the phase of signals with low-cost reflective elements. In this paper, we investigate the RIS-assisted D2D communication system under Nakagami-m fading. Overlay mode and underlay mode are discussed respectively. To characterize the performance of proposed communication system, we propose novel closed-form analytical expressions for outage probability under two modes, respectively. Based on the analytical results, we derive the diversity order of outage probability. We show that under overlay D2D mode, the diversity order depends on the number of elements and fading parameters. For the underlay D2D case, 0 diversity order is obtained due to the existence of interference. Furthermore, we study the ergodic achievable rate for each mode and the corresponding tight bounds are derived in …