Perturbed moments and a longer mollifier for critical zeros of $$\zeta $$ ζ

Abstract

Let A(s) be a general Dirichlet polynomial and be a smooth function supported in [1,2] with mild bounds on its derivatives. New main terms for the integralT dt are given. For the error term, we show that the length of the Feng mollifier can be increased from θ < 17 33 to θ < 6 11 by decomposing the error into Type I and Type II sums and then studying the resulting sums of Kloosterman sums. As an application, we slightly increase the proportion of zeros of ζ (s) on the critical line.