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