Primes with a missing digit: distribution in arithmetic progressions and an application in sieve theory

TL;DR

This paper establishes Bombieri-Vinogradov type theorems for primes missing a digit in their base-$b$ expansion, using the circle method, and applies these results to count primes of a specific quadratic form with missing digits.

Contribution

It introduces new distribution results for primes with missing digits in their base-$b$ expansion and applies these to sieve theory for primes of the form p=1+m^2+n^2.

Findings

Proves distribution theorems for primes with missing digits in arithmetic progressions.

Provides bounds on the number of primes of the form p=1+m^2+n^2 with missing digits.

Utilizes the circle method and semi-linear sieve techniques.

Abstract

We prove Bombieri-Vinogradov type theorems for primes with a missing digit in their $b$-adic expansion for some large positive integer $b$. The proof is based on the circle method, which relies on the Fourier structure of the integers with a missing digit and the exponential sums over primes in arithmetic progressions. Combining our results with the semi-linear sieve, we obtain an upper bound and a lower bound of the correct order of magnitude for the number of primes of the form $p=1+m^2+n^2$ with a missing digit in a large odd base $b$.