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

Abstract

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 : ℜ z∈(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 : ℜ z =1/2} ; and that 3) all the zeros of the Riemann zeta function located on the critical line are simple.