On the zeros of Riemann's zeta-function on the critical line

We combine the mollifier method with a zero detection method of Atkinson to prove in a new way that a positive proportion of the nontrivial zeros of the Riemann zeta-function ζ(s)ζ(s) are on the critical line. One of the main ingredients of the proof is an estimate for a mollified fourth moment of ζ(s)ζ(s). We deduce this estimate from the twisted fourth moment formula that has been recently developed by Hughes and Young.