Content caching is a technique aimed to reduce the network load imposed by data transmission during peak time while ensuring users' quality of experience. Studies have shown that content delivery via coded caching can significantly improve beyond the performance limits of conventional caching schemes when caches and the server share a common link. Finding the optimal cache content placement however, becomes challenging under arbitrary distributions of content popularity. While existing works show that partitioning contents into three popularity levels performs better when multiple requests are received at each time slot, they neither delve into the problem analysis nor derive closed-form expressions for the optimum partitioning problem. In this paper, we analyze the coded caching scheme for a system with arbitrary content popularity, where we derive explicit closed-forms for the server load in the delivery phase and formulate the near-optimum partitioning problem. Simulation results are presented to corroborate our mathematical analysis.