We introduce a new mollifier and apply the method of Levinson and Conrey to prove that at least 41.28% of the zeros of the Riemann zeta function are on the critical line. The method may also be used to improve other results on zeros relate to the Riemann zeta function, as well as conditional results on prime gaps.

Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 2.7327 times the average spacing and infinitely often they differ by at most 0.5154 times the average spacing.