Many applications concerning primes in arithmetic progressions are hindered by the possible existence of a real zero of a Dirichlet L-function. These could only occur when L is attached to a quadratic character χ mod q. We should like to rule out such zeroes for all ‘small’ moduli, say, all q < Q. (These margins are not wide enough to include a proof that we can take Q = ∞.) I will outline some recent work, joint with Dave Platt, that allows us to take a much larger Q than was previously known.