For a large class of digital functions $f$, we estimate the sums ${\sum }_{n\le x}\Lambda \left(n\right)f\left(n\right)$ (and ${\sum }_{n\le x}\mu \left(n\right)f\left(n\right)$, where $\Lambda$ denotes the von Mangoldt function (and $\mu$ the Möbius function). We deduce from these estimates a Prime Number Theorem (and a Möbius randomness principle) for sequences of integers with digit properties including the Rudin-Shapiro sequence and some of its generalizations.