The Distribution of the Nontrivial Zeros of Riemann Zeta Function

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 $\chi (\frac{1}{2}+it)$.

More detailed introduction and two figures are added in version 2