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Given an ergodic dynamical system $(X, \mathcal{B}, \mu, T)$, we prove thatfor each function $f$ belonging to the Orlicz space $L(\log L)^2(\log \logL)(X, \mu)$, the ergodic averages \[ \frac{1}{\pi(N)} \sum_{p \in \mathbb{P}_N}f\big(T^p x\big), \] converge for $\mu$-almost all $x \in X$, where$\mathbb{P}_N$ is the set of prime numbers not larger that $N$ and $\pi(N) = \#\mathbb{P}_N$.

Endpoint estimates for the maximal function over prime numbers