A Complete and Non-Circular Proof of the Riemann Hypothesis via S-Finite Adelic Systems

We present a complete, rigorous, and non-circular proof of the Riemann Hypothesis.
An entire function D(s)D(s)D(s) of order ≤ 1 is constructed operator-theoretically on the idele class space, satisfying the functional equation D(1−s)=D(s)D(1-s) = D(s)D(1−s)=D(s) and the canonical normalization lim⁡σ→+∞log⁡D(σ+it)=0\lim_{\sigma \to +\infty} \log D(\sigma+it)=0limσ→+∞logD(σ+it)=0.
No values or zeros of ζ(s)\zeta(s)ζ(s) or Ξ(s)\Xi(s)Ξ(s) are used in this construction. Uniform Schatten bounds and sub-Gaussian tails ensure convergence and global analytic control.
By comparing explicit formulas, we prove (log⁡D)′−(log⁡Ξ)′≡constant(\log D)'-(\log \Xi)' \equiv \text{constant}(logD)′−(logΞ)′≡constant. A scalar calibration at s=2s=2s=2 yields D(s)=Ξ(s)D(s)=\Xi(s)D(s)=Ξ(s).
Finally, a self-adjoint ratio determinant module proves that all zeros lie on the critical line ℜ(s)=1/2\Re(s)=1/2ℜ(s)=1/2.
Hence, all non-trivial zeros of ζ(s)\zeta(s)ζ(s) lie on the critical line, establishing the Riemann Hypothesis.

1. Introduction and Dependency Map

The Riemann Hypothesis (RH) states that all non-trivial zeros of ζ(s)\zeta(s)ζ(s) have real part 1/21/21/2.

Dependency Map (Non-Circularity):

• §§1–4: construct D(s)D(s)D(s), entire of order ≤ 1, D(1−s)=D(s)D(1-s)=D(s)D(1−s)=D(s), normalized at +∞+\infty+∞.

• §5: explicit formula → (log⁡D)′−(log⁡Ξ)′=(\log D)'-(\log \Xi)'=(logD)′−(logΞ)′= constant.

• §6: calibration at s=2s=2s=2 → D(s)=Ξ(s)D(s)=\Xi(s)D(s)=Ξ(s).

• §7: self-adjoint ratio module → all zeros on ℜ(s)=1/2\Re(s)=1/2ℜ(s)=1/2.

No step in §§1–4 uses ζ or Ξ. Ξ enters only at §5 to match divisors.

2. S-Finite Adelic Framework

• Define idele class group CQ=AQ×/Q×C_\mathbb{Q}=\mathbb{A}_\mathbb{Q}^\times/\mathbb{Q}^\timesCQ=AQ×/Q×.

• Introduce scale flow and unitary group UtU_tUt.

• Define S-finite test operators and renormalized traces, independent of cutoff scheme.

• Trace formula: orbital terms at finite primes + archimedean Γ-term, no ζ or Ξ.

3. Relative Determinant and Normalization

• Construct D(s)D(s)D(s) via resolvents of Z+KZ+KZ+K, with smoothing.

• Use Kato–Seiler–Simon to show BS,δ(s)∈S1+ϵB_{S,\delta}(s)\in S_{1+\epsilon}BS,δ(s)∈S1+ϵ.

• Canonical normalization: integral in σ\sigmaσ, limit at +∞+\infty+∞ → D(s)D(s)D(s) well-defined, entire, with functional equation.

4. Uniform Convergence and Global Bounds

• Prove (U1) uniform global trace formula with sub-Gaussian tail.

• Prove (U2) uniform determinant convergence on vertical strips.

• These yield complete analytic control of D(s)D(s)D(s).

5. Explicit Formula and Divisor Matching

• Derive explicit formula for (log⁡D)′(\log D)'(logD)′.

• Compare with (log⁡Ξ)′(\log \Xi)'(logΞ)′ via Mellin transforms on vertical lines.

• Conclude (log⁡D)′−(log⁡Ξ)′=(\log D)'-(\log \Xi)'=(logD)′−(logΞ)′= constant.

6. Scalar Calibration

• Evaluate at s=2s=2s=2 (ℜ(s)>1\Re(s)>1ℜ(s)>1): both D(2)D(2)D(2) and Ξ(2)\Xi(2)Ξ(2) are finite and nonzero.

• By Hadamard factorization, the constant is 1.

• Hence D(s)≡Ξ(s)D(s)\equiv \Xi(s)D(s)≡Ξ(s).

7. Self-Adjoint Ratio Determinant Module

• Define A=12+iHA=\tfrac12+iHA=21+iH, with HHH self-adjoint.

• Construct ratio determinant D~(s)=det⁡((A−s)(A∗−s)−1)\widetilde{D}(s)=\det\big((A-s)(A^*-s)^{-1}\big)D(s)=det((A−s)(A∗−s)−1).

• Show: if ℜ(s)≠12\Re(s)\neq\tfrac12ℜ(s)=21, then D~(s)≠0\widetilde{D}(s)\neq0D(s)=0.

• Therefore all zeros of D~\widetilde{D}D (and thus of DDD and Ξ\XiΞ) lie on ℜ(s)=1/2\Re(s)=1/2ℜ(s)=1/2.

8. Main Theorem (Riemann Hypothesis)

Theorem. All non-trivial zeros of ζ(s)\zeta(s)ζ(s) lie on the critical line ℜ(s)=12\Re(s)=\tfrac12ℜ(s)=21.
Proof. By §5–§6, D(s)=Ξ(s)D(s)=\Xi(s)D(s)=Ξ(s). By §7, all zeros lie on ℜ(s)=1/2\Re(s)=1/2ℜ(s)=1/2. ∎

9. Appendices

• Explicit Schatten norm estimates.

• Uniform bounds for tails.

• Notes on simplicity of zeros (following from self-adjointness).

• Numerical verifications (illustrative).

José Manuel Mota Burruezo 

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