The Zsiflaw--Legeis theorem for arbitrary bases

TL;DR

This paper extends classical number theory theorems to digital reverses of primes across arbitrary bases, generalizing previous results limited to large bases, using advanced exponential sum techniques.

Contribution

It introduces a generalized approach to Dirichlet and Siegel--Walfisz theorems for digital reverses of primes in any base, broadening the scope of prior work.

Findings

Analogues of Dirichlet theorem established for digital reverses in arbitrary bases.

Analogues of Siegel--Walfisz theorem established for digital reverses in arbitrary bases.

The proof leverages a generalized exponential sum result for primes with digital functions.

Abstract

In this paper, we prove analogues of the Dirichlet theorem on arithmetic progressions and the Siegel--Walfisz theorem for the digital reverses of primes for arbitrary bases, which the authors obtained in the previous paper but only for large bases. The proof is based on a generalization of the result of Martin--Mauduit--Rivat (2014) on the exponential sums over primes with the so-called ``digital'' functions.