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We investigate the influences of the electron-phonon interaction on the transport properties of one quantum-dot system with a side-coupled Majorana zero mode (MZM). Our calculation results show that at the zero-temperature limit, the MZM-governed zero-bias conductance value can be magnified, dependent on the interplay between electron-phonon interaction and dot-MZM coupling. In the case of finite temperature, the electron-phonon interaction makes leading contributions to the suppression of the magnitude of zero-bias conductance. We believe that this work can be helpful for understanding the signature of the MZM in electron transport through mesoscopic circuits.

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