With the increasing number of users and the emergence of different types of network services, a multi-server architecture has emerged in recent years. In order to ensure the secure communication of Internet participants in an open network environment, the authentication and key agreement protocol for multi-server architectures were proposed in the past. In 2018, Chatterjee et al. put forward a lightweight three-factor authentication and key agreement protocol for a multi-server environment, and they claimed that all known security features with satisfactory performance could be realized in their protocol. However, it is found that their scheme is vulnerable to user impersonation attacks and cannot achieve user un-traceability and three-factor security through our cryptanalysis. In order to solve these shortcomings, we propose a new lightweight and anonymous three-factor authentication scheme for the multi-server environment in this article. Furthermore, the proposed protocol is proved to be AKE secure theoretically, and we use BAN-logic to prove that our protocol realizes mutual authentication between communication participants. Finally, we show that our proposed scheme is practical and efficient through the comparison of security features and performance.