The Distribution of the Nontrivial Zeros of Riemann Zeta Function

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 $\mathrm{\partial }{D}_{ϵ}^1$ and $\mathrm{\partial }{D}_{ϵ}^2$ in the $s$ coordinate are specified, and improper logarithmic integrals of nonvanishing $\zeta (s)$ along $\partial D_{\epsilon}^1$and$\partial D_{\epsilon}^2$canbecalculatedas$2\pi i$and$0$respectively,dependingonthetotalincreaseintheargumentof$z=\chi(s)$\mathrm{.}Anydomainsinthecriticalstripforsufficientlylarge$t$ can be covered by the domains $D_{ϵ}^1$ or $D_{ϵ}^2$, 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 $\chi (\frac{1}{2}+it)$.