Take the Riemann zeta-function, ζ(s)=1+2−s+3−s+…ζ(s)=1+2−s+3−s+…, which converges whenever R(s)>1ℜ(s)>1. Now chop the series after NN terms and call the finite piece ζN(s)ζN(s). In 1948 Turan made the striking observation that if for sufficiently large NN the functions ζN(s)ζN(s) did not vanish for R(s)>1ℜ(s)>1, then the Riemann hypothesis would be true. In 1983 Montgomery showed that this is false: for all sufficiently large NN there are such zeroes. I shall discuss this problem and recent work with Dave Platt (at Bristol), that finally resolves the values of NN for which zeroes do, and do not, exist.