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

## TL;DR

This paper develops new main terms for integrals involving the Riemann zeta function and Dirichlet polynomials, extends mollifier length, and improves the lower bound on the proportion of zeros on the critical line.

## Contribution

It introduces refined main terms for zeta integrals and extends mollifier length from 17/33 to 6/11, enhancing zero proportion results.

## Key findings

- Extended mollifier length from 17/33 to 6/11.
- Improved main term formulas for zeta integrals.
- Slight increase in the known proportion of zeros on the critical line.

## Abstract

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