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 $G(z)=zeta(s)=zetacirctau(z)$ 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 $2pi i$ 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 Riemanns initial hypothesis and in rhythm with the argument of $chi(frac{1}{2}+it)$.