An attempt to prove Riemann

Th is an attempt to formally prove Riemann’s hypothesis (RH) related to the non-trivial zeros of the Riemann- Euler Zeta function, which are postulated by the hypothesis to lie on the critical line σ = 1/2. This paper depends on chapter 1 and chapter 2 from H.M. Edwards’ book “Riemann Zeta function” and it provides three different proofs of RH, all these proofs start from the somehow modified Xi function ξ(x), which is defined based on the functional equation of the Zeta function, separates it into real u(σ,t) and imaginary v(σ,t) parts, then equates both of them to zero and investigates the behavior of the resulting pair of equations u(σ,t) = 0 and v(σ,t) =0. The first proof investigates the algebraic and geometric structure of the Zeta function non-trivial zeros as dictated by the functional equation, using fundamental concepts from complex algebra and analytic geometry usually taught in high schools, the second proof simply depends on studying the validity of equality relation in the pair of equations u(σ,t) = 0 and v(σ,t) =0 in certain regions covering the whole plane and focusing on the critical strip. These regions are constructed based on a hyperbola relating σ and t and it results from rationalization of the equation u(σ,t) = 0. The last proof utilizes fundamental concepts from infinite series and products of trigonometric and hyperbolic functions to prove Riemann’s hypothesis.