Critical zeros of the Riemann zeta-function

In this unpublished note, we sketch an idea of using a three-piece mollifier to slightly improve the known percentages of zeros and simple zeros of the Riemann zeta-function on the critical line. This uses the recent result of Bettin, Bui, Li and Radziwill [BBLR] on the twisted fourth moment of the Riemann zeta-function.