We consider the PageRank problem in the dynamic setting, where the goal is to explicitly maintain an approximate PageRank vector $π\in \mathbb{R}^n$ for a graph under a sequence of edge insertions and deletions. Our main result is a complete characterization of the complexity of dynamic PageRank maintenance for both multiplicative and additive ($L_1$) approximations. First, we establish matching lower and upper bounds for maintaining additive approximate PageRank in both incremental and decremental settings. In particular, we demonstrate that in the worst-case $(1/α)^{Θ(\log \log n)}$ update time is necessary and sufficient for this problem, where $α$ is the desired additive approximation. On the other hand, we demonstrate that the commonly employed ForwardPush approach performs substantially worse than this optimal runtime. Specifically, we show that ForwardPush requires $Ω(n^{1-δ})$ time per update on average, for any $δ> 0$, even in the incremental setting. For multiplicative approximations, however, we demonstrate that the situation is significantly more challenging. Specifically, we prove that any algorithm that explicitly maintains a constant factor multiplicative approximation of the PageRank vector of a directed graph must have amortized update time $Ω(n^{1-δ})$, for any $δ> 0$, even in the incremental setting, thereby resolving a 13-year old open question of Bahmani et al.~(VLDB 2010). This sharply contrasts with the undirected setting, where we show that $\rm{poly}\ \log n$ update time is feasible, even in the fully dynamic setting under oblivious adversary.