Prime numbers along Rudin–Shapiro sequences

Abstract

For a large class of digital functions $f$, we estimate the sums ${\sum }_{n\le x}\Lambda (n)f(n)$ (and ${\sum }_{n\le x}\mu (n)f(n)$, 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.