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

We prove Bombieri-Vinogradov type theorems for primes with a missing digit in their b$b$-adic expansion for some large positive integer b$b$. 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+m2+n2$p=1+m<^>2+n<^>2$ with a missing digit in a large odd base b$b$.