All Zeros of the Riemann Zeta Function in the Critical Strip Are Located on the Critical Line and Are Simple  

作　　者：Frank Stenger Frank Stenger(University of Utah, Salt Lake City, USA)

机构地区：[1]University of Utah, Salt Lake City, USA

出　　处：《Advances in Pure Mathematics》2023年第6期402-411,共10页理论数学进展（英文）

摘　　要：In this paper we study the function , for z∈C. We derive a functional equation that relates G(z) and G(1−z) for all z∈C, and we prove: 1) that G and the Riemann zeta function ζ have exactly the same zeros in the critical region D:= {z∈C:ℜz∈(0,1)};2) the Riemann hypothesis, i.e., that all of the zeros of G in D are located on the critical line := {z∈D:ℜz =1/2};and that 3) all the zeros of the Riemann zeta function located on the critical line are simple.In this paper we study the function , for z∈C. We derive a functional equation that relates G(z) and G(1−z) for all z∈C, and we prove: 1) that G and the Riemann zeta function ζ have exactly the same zeros in the critical region D:= {z∈C:ℜz∈(0,1)};2) the Riemann hypothesis, i.e., that all of the zeros of G in D are located on the critical line := {z∈D:ℜz =1/2};and that 3) all the zeros of the Riemann zeta function located on the critical line are simple.

关 键 词：Riemann Hypothesis Fourier Transforms Schwarz Reflection Principle Cauchy-Riemann Equations Trapezoidal-Midordinate Quadrature 

分 类 号：O17[理学—数学]