A Spectral-Geometric Proof of the Riemann Hypothesis

Abstract

We establish the following result: every non-trivial zero of the Riemann zeta-function ζ(s), lies on the critical line Re (s) = 1/2. Following the approach suggested by the Hilbert–Pólya conjecture, We presents a complete and rigorous proof of the Riemann hypothesis. We first construct a specific one-dimensional, compact, boundaryless, smooth circle manifold S¹ . We then impose even-type doubling, the symmetry and minimal nontrivial requirement to constrain a self-adjoint elliptic operator acting on this manifold,

D(z) = D₁ + qi(z - b)Γ

Using the Hermiticity of this operator, we derive the reflection identity

Z(z) = (Z(1 - z̄))̄ = det_{ζ,θ}[D(z)] = 0

Analyzing the structure of  D(z) , we conclude that  z₁ = z and z₂ = 1 - z̄  lie on the same kernel line, i.e.,

ker D(z₁) = ker D(z₂) , R(z) = b

Finally, we obtain the equation of  and the Riemann -function  i.e.,

Z(Φ(s))=ξ(s),  z=Φ(s)=b+ie^(((s-a))/i),

thereby establishing that all nontrivial zeros of  lie on the critical line Re s = 1/2. 

Keywords:Riemann Hypothesis; Dirac Operator; Heat Kernel; Hamburger Theorem; Riemann–von Mangoldt Formula