• DOI:10.1007/s40687-019-0199-8
• Corpus ID: 202542332

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

@article{Pratt2018MoreTF,
  title={More than five-twelfths of the zeros of \$\$\zeta \$\$ are on the critical line},
  author={Kyle Pratt and Nicolas Robles and Alexandru Zaharescu and Dirk Zeindler},
  journal={arXiv: Number Theory},
  year={2018}
}

• Kyle Pratt, Nicolas Robles, +1 author D. Zeindler
• Published 28 February 2018
• Mathematics
• arXiv: Number Theory

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.