The Distribution of the Nontrivial Zeros of Riemann Zeta Function [article]

Jianyun Zhang
2020 arXiv   pre-print
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_-(1/z) in the z coordinate, where G(z)=ζ(s)=ζ∘τ(z) and G_- is the
more » ... gated branch of G. Two types of special and symmetric contours ∂ D_ϵ^1 and ∂ D_ϵ^2 in the s coordinate are specified, and improper logarithmic integrals of nonvanishing ζ(s) along ∂ D_ϵ^1 and ∂ D_ϵ^2 can be calculated as 2π i 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(1/2+it).
arXiv:2005.04568v2 fatcat:6rkc5hbpmrduvjomok6gcajss4