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

Abstract

AbstractWe prove Bombieri–Vinogradov type theorems for primes with a missing digit in their ‐adic expansion for some large positive integer . 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 with a missing digit in a large odd base .