Prime numbers along Rudin–Shapiro sequences

Christian Mauduit (1) , Joël Rivat (1)

1 I2M - Institut de Mathématiques de Marseille

For a large class of digital functions f, we estimate the sums Sigma(n <= x) Lambda(n)f (n) (and Sigma(n <= x) mu(n)f (n)), where Lambda denotes the von Mangoldt function (and mu, the Mobius function). We deduce from these estimates a Prime Number Theorem (and a Mobius randomness principle) for sequences of integers with digit properties including the Rudin-Shapiro sequence and some of its generalizations.

Mots clés: Rudin–Shapiro sequence; prime numbers; Möbius function; exponential sums

Domaines: Théorie des nombres [math.NT]

Christian Mauduit, Joël Rivat. Prime numbers along Rudin–Shapiro sequences. Journal of the European Mathematical Society, 2015, 17 (10), pp.2595-2642. ⟨10.4171/JEMS/566⟩. ⟨hal-01259940⟩