  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.
