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

15 Sep 2019 Pratt Kyle Robles Nicolas Zaharescu Alexandru Zeindler Dirk

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... (read more)

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  • NUMBER THEORY