The Littlewood-Paley formula and mean counting function for vertical limits of Dirichlet series

We prove a Littlewood-Paley formula for the Hardy space of Dirichlet series $\mathscr{H}^p$ with $1\leq p<\infty$ in terms of almost every vertical limit function. This significantly strengthens previous results, which hold either only as an average over the vertical limit functions or under additional assumptions of uniform convergence. As part of our approach, we obtain a Hardy-Stein identity for the derivative of the $p$-mean of almost every vertical limit. We further show that the mean counting function exists for any $f$ in $\mathscr{H}^p$ in terms of almost all of its vertical limit functions. This is done by establishing a version of Jensen's formula in this setting. In the process, we also deduce ergodic versions of Fatou's lemma and the monotone and dominated convergence theorems for the Kronecker flow.