The sequential hypothesis testing technique has been broadly applied in a large number of engineering applications because it provides the efficient rules to make a decision of accepting one of hypotheses at any stage of the experiment. One important application is the optimal caching scheme design in 5G and beyond mobile wireless networks through observing the data contents requested by mobile users as an experiment and making a decision on what distribution of the requested data-content is. Making this decision, cache stations in mobile wireless network are able to estimate the popularity of requested data contents in the future and thus proactively cache the popular data contents in nearby mobile users to avoid the retransmission and its delay for the same data, optimizing the time-sensitive data download services. Towards this end, we model the optimal caching as the estimation strategy problem of future data popularity through developing the optimal stopping and decision rules under the Zipf sequential hypothesis testing. First, we show that our developed Zipf sequential hypothesis testing is exponentially bounded. Then, we derive the lower bound of the stopping time. Finally, we derive the closed-form solution of optimal stopping and decision rules for sequential hypothesis testing under the Zipf distribution.