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TL;DR

This paper extends a theorem on the Rudin-Shapiro sequence to a broader class of digit-based functions, establishing a prime number theorem for these sequences and addressing a question by Kalai.

Contribution

It generalizes Mauduit and Rivat's theorem by weakening hypotheses, proving a prime number theorem for generalized Rudin-Shapiro and block-additive sequences.

Findings

Proves a prime number theorem for a large class of digit-defined functions

Includes generalized Rudin-Shapiro sequences and block-additive sequences

Provides a partial answer to Kalai's question

Abstract

In this paper, we generalize Mauduit and Rivat's theorem on the Rudin-Shapiro sequence. Weakening the hypothesis needed in their theorem, we prove a prime number theorem for a large class of functions defined on the digits. Our result covers the case of generalized Rudin-Shapiro sequences as well as bloc-additive sequences on finite and infinite expansions. We also give a partial answer to a question posed by Kalai.