optimal action-value function

The optimal action-value function $q_{\ast }(s,a)$ is the maximum action-value function over all policies. In other words, $q_{\ast }(s)$ is the resulting action-value function when the agent follows the optimal policy (treat $\pi$ as a parameter in the equation below).

$$q_{\ast }(s,a)=\underset{\pi }{\max }{q}_{\pi }(s,a)$$