Twisted second moments of the Riemann zeta-function and applications

In order to compute a twisted second moment of the Riemann zeta-function, two different mollifiers, each being a combination of two different Dirichlet polynomials, were introduced separately by Bui, Conrey, and Young, and by Feng. In this article we introduce a mollifier which is a combination of four Dirichlet polynomials of different shapes. We provide an asymptotic result for the twisted second moment of ζ(s) for such choice of mollifier. A small increment on the percentage of zeros of the Riemann zeta-function on the critical line is given as an application of our results.