Blocs de chiffres de taille croissante dans les nombres premiers

TL;DR

This paper proves a theorem related to digital blocks in prime numbers, extending previous results to sequences with growing block lengths that are not automatic, using harmonic analysis and refined carry propagation techniques.

Contribution

It introduces new methods to analyze digital blocks in prime numbers with increasing length, expanding the scope of prior prime number theorems.

Findings

Established a prime number theorem for sequences with growing digital blocks

Controlled sums of type I and II for non-automatic sequences

Developed refined carry propagation techniques

Abstract

In this article, we prove a theorem \`a la Mauduit et Rivat (prime number theorem, Moebius randomness principle) for functions that count digital blocks whose length is a growing function tending to infinity. These sequences are not automatic. To obtain our results, we control sums of type I and II and use an adapted and refined version of the carry propagation property as well as standard methods from harmonic analysis.