# Endpoint estimates for the maximal function over prime numbers

## TL;DR

This paper establishes almost sure convergence of ergodic averages over prime numbers for functions in a specific Orlicz space, extending classical results to a new setting involving primes and advanced function spaces.

## Contribution

It proves endpoint estimates for the maximal function over primes in the context of ergodic theory, particularly for functions in the Orlicz space $L( ext{log} L)^2( ext{log} ext{log} L)$.

## Key findings

- Almost sure convergence of ergodic averages over primes for functions in the specified Orlicz space.
- Extension of classical ergodic theorems to prime number averages with endpoint estimates.
- New techniques for handling maximal functions over primes in ergodic theory.

## Abstract

Given an ergodic dynamical system $(X, \mathcal{B}, \mu, T)$, we prove that for each function $f$ belonging to the Orlicz space $L(\log L)^2(\log \log L)(X, \mu)$, 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 $\pi(N) = \# \mathbb{P}_N$.