Proof of the Riemann Hypothesis, Schemes and Mind Maps of Applied Mathematics

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A Complete Proof to

the Riemann

Hypothesis

A Supplementary to the paper titled “A Duality to the Birch & Swinnerton-Dyer Conjecture and the Visualized Sandwich Proof to the Riemann Hypothesis

Immediate Value

Theorem

Rolle’s Theorem

On the contrary, assume that there are another critical

line s’ in the critical region 0 < R(s') ≠ 0.5 <1 such

that ξ(Re(s’) + Iy) = 0 which has all of the same

properties as ξ(0.5 + Iy) = 0. In practice, let s’ = Re (s’)

+ Iy. Then there must be a line like x’ = Re(s’) such

that the line x’ must contain the point s’ = Re(s’) + Iy.

(N.B. In the present proof, we authors use Re(s’) = 0.

or Re(s’) = 0.7 as the case studies in the following

discussion.)

( εz, εy ) = d (R (ζ ( x' ± δ x ± Iy ± δy )), R (ζ (R ( s' ) ± δx ± Iy ))), there will always be a ( δx, δy ) with (δx, δy) = (R (ζ-1 (± εZ+R (ζ (x' - R ( s ') ± Iy ))),(± εy + ( Iy - Iy )))) where (δx, δy) → (| ( x'- R ( s' ))|, |( y - y )|) when ( εZ, εy ) → (0,0). and also 2 im ik in e e π π± θ - x for n = 0,1,2…,n-1 lies on the upper line Re(ζ ( x’ ± δx+Iy )) = εz and the lower line Re(ζ ( x’ ± δx+Iy )) = -εz with width (-εz, εz) → (0,0) and sandwiched to approaching the middle critical line x = 0.5. Or 0 = -εz = - Re(ζ ( x’± δx+Iy )) .≤. Re(ζ (Re( s’ ) + Iy )) .≤. Re(ζ ( x’ ± δx +Iy )) = εz = 0 Then Re(ζ (Re ( s ’) + Iy )) = 0 which is just the definition of continuous.

By the Immediate Value Theorem, if the curve f(x) is continuous in a closed interval such as [a,b] s.t. f(a) < 0 and f(b) > 0, then there must be a point “c”, staying between a & b and f(c) = 0

Real & Imaginary Parts of the ζ(0.1+It) Real & Imaginary Parts of the ζ(0.5+It) Real & Imaginary Parts of the ζ(0.7+I*t)

1. Point on a line and line contains

the point;

2. Positive and negative (outward

and backward) angular rotation;

3. Sandwich Theorem implies

continuous;

4. Immediate Value Theorem implies

x = 0.5 is the optimal root;

5. normal non-trivial zeta zeros =

abnormal non-trivial zeta zeros &

Riemann Hypothesis is true.

A Summary for the Proof of the Riemann Hypothesis

Conclusion(s)

There may be other zeros or the intersection points at x = 0.1 or x = 0.7. But we have shown in my present reference titled paper[1], these intersection points are actually the intersection points of the critical line x = 0.5 which also contains the optimal root. We (Lam & Siu) have (visually) proved that the Riemann Hypothesis is actually correct.