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

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 $b$.

Accepted version, fixed a few typos, rewrote parts III and IV, a small necessary fix in the proof of Lemma 6.2 was applied

References in corpus (2)

• The Goldbach Problem for Primes That Are Sums of Two Squares Plus One
• Primes from sums of two squares and missing digits