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

Abstract

Let $A(s)$ be a general Dirichlet polynomial and $Φ$ be a smooth function supported in $[1,2]$ with mild bounds on its derivatives. New main terms for the integral $I(α,β)=\int_{\mathbb{R}} ζ(\frac{1}{2}+α+it)ζ(\frac{1}{2}+β+it)|A(\frac{1}{2}+it)|^2 Φ(\frac{t}{T})dt$ are given. For the error term, we show that the length of the Feng mollifier can be increased from $θ< \frac{17}{33}$ to $θ< \frac{6}{11}$ 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.