The Distribution of the Nontrivial Zeros of Riemann Zeta Function (2005.04568v2)

Abstract: We improve the estimation of the distribution of the nontrivial zeros of Riemann zeta function $\zeta(\sigma+it)$ for sufficiently large $t$, which is based on an exact calculation of some special logarithmic integrals of nonvanishing $\zeta(\sigma+it)$ along well-chosen contours. A special and single-valued coordinate transformation $s=\tau(z)$ is chosen as the inverse of $z=\chi(s)$, and the functional equation $\zeta(s) = \chi(s)\zeta(1-s)$ is simplified as $G(z) = z\, G_-(\frac{1}{z})$ in the $z$ coordinate, where $G(z)=\zeta(s)=\zeta\circ\tau(z)$ and $G_-$ is the conjugated branch of $G$. Two types of special and symmetric contours $\partial D_{\epsilon}^1$ and $\partial D_{\epsilon}^2$ in the $s$ coordinate are specified, and improper logarithmic integrals of nonvanishing $\zeta(s)$ along $\partial D_{\epsilon}^1$ and $\partial D_{\epsilon}^2$ can be calculated as $2\pi i$ and $0$ respectively, depending on the total increase in the argument of $z=\chi(s)$. Any domains in the critical strip for sufficiently large $t$ can be covered by the domains $D_{\epsilon}^1$ or $D_{\epsilon}^2$, and the distribution of nontrivial zeros of $\zeta(s)$ is revealed in the end, which is more subtle than Riemann's initial hypothesis and in rhythm with the argument of $\chi(\frac{1}{2}+it)$.