Riemann’s zeta函数零点分布的几个性质

The Properties of the Zero Points Distribution for the Riemann's Zeta Function

表示Riemann’s zeta函数ζ(z)在半直线{Re(z)=1/2,Im(z)】0}上的零点。

Let τ_γ=1/2+iβ_γ=|τ_γe^(iα),0<α_γ<π/2,β_γ>0,γ=1,2,… denote the zreo points of the Riemenn's function ζ(z), Lying on semi-line {Re(z) =1/2,Im(z)>0}.The main results of the paper aer as following: Theorem 1. Let ζ_0(z) =ζ(z+1/2)/ζ(1/2), then, a necessary and suffictent condition for the truth of thd Riemann hypothesis is where, ξ is Euler's constant. Theorom 2.A necessary and sufficient condition for the truth of the Riemann hypothesisis Theorem 3.Anecessary and sufficient condition for the truth of the Riemann hypothesisis to exist a positive integer q(≥z) such that the function ξ(z) has not zero points in the domein {Re(z)>1/2, 0≤arg z≤π/2 (1-1/q)}, and hold the following identity Theorem 4. A necessary and sufficient condition for the truth of the Riemann hypoth- esis is to exist a positive integer n such that the function ζ(z) has not zero points in the do- main {0≤arg(z-1/2)≤π/2(1-1/(2n))},and hold the following identity