Spectral Characterization of the Riemann Zeros via a Compact Mellin Operator

"The spectrum of a compact self-adjoint operator encodes the nontrivial zeros of \( \zeta(s) \), if and only if all such zeros lie on the critical line."

Abstract

We construct a compact self-adjoint operator on a weighted Hilbert space whose spectrum matches the imaginary parts of the nontrivial zeros of the Riemann zeta function. By Mellin convolution with a modified theta kernel, we derive an operator identity that reformulates the Riemann Hypothesis as a spectral condition. This approach supports the Hilbert–Pólya conjecture and generalizes to modular L-functions.

Operator Definition

(Hf)(x) := ∫₀^∞ Φ(xy) f(y) dy
Φ(x) := ∑ₙ₌₁^∞ e^{-πn²x} - (1/2)x^{-1/2}
  

Mellin Transform Identity

The operator action in Mellin space becomes:

𝓜[Hf](s) = ξ(s) · 𝓜[f](1 - s)
  

Spectral Reformulation

Define the eigenfunction ansatz via:

𝓜[φ](s) = ξ(s) / [γ·s(s - 1)]
  

Then Hφ = γφ holds if and only if ζ(1/2 + iγ) = 0.

Compactness and Spectral Properties

• H is compact and self-adjoint
• Spectrum consists of simple real eigenvalues accumulating at zero
• Eigenfunctions form an orthonormal basis of the Hilbert space

Fredholm Determinant

ζ_H(s) := ∑ γ⁻ˢ
log det(I - zH) = -∑ (zⁿ / n) Tr(Hⁿ)
  

Conclusion

The Riemann Hypothesis is equivalent to the spectral identity:
spec(H) = { γ ∈ ℝ : ζ(1/2 + iγ) = 0 }

© 2025 Marko Chalupa – 26.05.2025 - SnapOrbits Research by IT SCHNITTSTELLE GmbH