100% of the zeros of the Riemann zeta-function are on the critical line (1805.07741v6)

Abstract: We consider a specific family of analytic functions $g_{\alpha,T}(s)$, satisfying certain functional equations and approximating to linear combinations of the Riemann zeta-function and its derivatives of the form $c_0\zeta(s)+c_1\frac{\zeta'(s)}{\log T}+c_2\frac{\zeta''(s)}{(\log T)^2}+\dots+c_{K}\frac{\zeta^{(K)}(s)}{(\log T)^{K}}$. We also consider specific mollifiers of the form $M(s)D(s)$ for these linear combinations, where $M(s)$ is the classical mollifier, that is, a short Dirichlet polynomial for $1/\zeta(s)$, and the Dirichlet polynomial $D(s)$ is also short but with large and irregular Dirichlet coefficients, and arises from substitution for $w$, in Runge's complex approximation polynomial for $f(w)=\frac1{c_0+w}$, of the Selberg approximation for $\frac{c_1}{\log T}\frac{\zeta'}{\zeta}(s)+\frac{c_2}{(\log T)^2}\frac{\zeta''}{\zeta}(s)+\dots+\frac{c_{K}}{(\log T)^{K}}\frac{\zeta^{(K)}}{\zeta}(s)$ (analogous to Selberg's classical approximation for $\frac{\zeta'}{\zeta}(s)$). Exploiting the functional equations previously mentioned (concerning translation of the variable $s$), together with the mean-square asymptotics of the Levinson-Conrey method and the Selberg approximation theory (with some additional results) we show that almost all of the zeros of the Riemann zeta-function are on the critical line.