An attempt to prove Riemann

02 August 2024, Version 1
This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.

Abstract

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.

Keywords

Analytic function
Cauchy-Riemann equations
Conic sections
Critical line
Critical strip
Eta function η(x)
Functional equation
Gamma function Г(x)
Hyperbola
Integration mean value theorem
Jacobi theta function \vartheta(x)
Non-trivial zero
Pole
Product formula
Prove by contradiction
Psi function ψ(x)
Riemann hypothesis(RH)
Riemann-Euler Zeta function ζ(s)
Integral second mean value theory-SMVT
Trivial zero
Xi Function ξ(x)
Zero (of a function).

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Comment number 1, Arshed Basheer: Aug 21, 2024, 11:26

I'll continue on each of these proofs separately, in the next paper I will complete the first proof by showing that taking any other non-trivial zero of the Riemann-Euler Zeta function (which will automatically result in three additional non-trivial zeros directly from the functional equation) will construct a new rectangle in the plane, and taking a third non-trivial zero will construct a third rectangle and do on. All these rectangles are symmetric with respect to the critical line (σ =1/2) and the real axis (t=0) and this in return will require all rectangles such constructed to have the same two diagonals and the same center (point at which the two diagonals intersect) and this common center at (1/2,0). This implies that the non-trivial zeros should be collinear in pairs which is to say they should all EITHER lie on these two inclined lines OR all lie on the critical line and the former case which occurs when σ not equal 1/2 will always lead to a solution (if it exits) that require t to be bounded thus violating the functional equation. This approach is simple and intuitive and can easily be proved using elementary plane geometry. The second proof will also be continued by showing that the two parameters c and d resulting from application of SMVT of integral are equal and this will result in a simplified transcendental equation to determine non-trivial zeros and will enable us to apply the argument principle to find a formula for the number of non-trivial zeros in (0,T).