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

Abstract

The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form $(μ\star Λ_1^{\star k_1} \star Λ_2^{\star k_2} \star \cdots \star Λ_d^{\star k_d})$ is computed unconditionally by means of the autocorrelation of ratios of $ζ$ 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 \[ ζ(s) + λ_1 \frac{ζ'(s)}{\log T} + λ_2 \frac{ζ''(s)}{\log^2 T} + \cdots + λ_d \frac{ζ^{(d)}(s)}{\log^d T}, \] where $ζ^{(k)}$ stands for the $k$th derivative of the Riemann zeta-function and $\{λ_k\}_{k=1}^d$ 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.