Primes with restricted digits

TL;DR

This paper proves there are infinitely many primes missing a specific digit in their decimal expansion by applying advanced analytic number theory techniques, including the circle method and Harman's sieve, to a digit-restricted prime problem.

Contribution

It introduces a novel combination of the circle method, harmonic analysis, and sieve techniques to address primes with digit restrictions, establishing their infinitude.

Findings

Infinitely many primes omit a given digit in decimal expansion.

Development of new estimates for Fourier transforms of digit-restricted numbers.

Application of harmonic analysis and sieve methods to digit-restricted prime problems.

Abstract

Let $a_0\in\{0,\dots,9\}$. We show there are infinitely many prime numbers which do not have the digit $a_0$ in their decimal expansion. The proof is an application of the Hardy-Littlewood circle method to a binary problem, and rests on obtaining suitable `Type I' and `Type II' arithmetic information for use in Harman's sieve to control the minor arcs. This is obtained by decorrelating Diophantine conditions which dictate when the Fourier transform of the primes is large from digital conditions which dictate when the Fourier transform of numbers with restricted digits is large. These estimates rely on a combination of the geometry of numbers, the large sieve and moment estimates obtained by comparison with a Markov process.