More than five-twelfths of the zeros of $\zeta$ are on the critical line

TL;DR

This paper computes the second moment of the Riemann zeta-function twisted by specific Dirichlet polynomials, leading to an improved lower bound of over five-twelfths for the zeros on the critical line.

Contribution

It introduces an unconditional calculation of the second moment with new techniques, enhancing the understanding of mollification and increasing the known lower bound of zeros on the critical line.

Findings

Unconditional second moment calculation of twisted zeta-functions

Description of the mollification process for zeta and its derivatives

Lower bound of zeros on the critical line exceeds five-twelfths

Abstract

The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form $(\mu \star \Lambda_1^{\star k_1} \star \Lambda_2^{\star k_2} \star \cdots \star \Lambda_d^{\star k_d})$ is computed unconditionally by means of the autocorrelation of ratios of $\zeta$ techniques from Conrey, Farmer, Keating, Rubinstein and Snaith (2005), Conrey, Farmer and Zirnbauer (2008) as well as Conrey and Snaith (2007). This in turn allows us to describe the combinatorial process behind the mollification of \[ \zeta(s) + \lambda_1 \frac{\zeta'(s)}{\log T} + \lambda_2 \frac{\zeta''(s)}{\log^2 T} + \cdots + \lambda_d \frac{\zeta^{(d)}(s)}{\log^d T}, \] where $\zeta^{(k)}$ stands for the $k$th derivative of the Riemann zeta-function and $\{\lambda_k\}_{k=1}^d$ are real numbers. Improving on recent results on long mollifiers and sums of Kloosterman sums due to Pratt and Robles (2017), as an application, we increase the current lower bound of critical zeros of the Riemann zeta-function to slightly over five-twelfths.