Scattering-Zeta Reconstruction

Abstract

All shifted normalized modular-scattering readings reconstruct the Riemann zeta value.

Theorem 1.1 (The shifted scattering products converge to zeta).

$$\forall z\in \mathbb{C},\Re (z)>1\Rightarrow \underset{N\to \infty }{\lim }\prod _{j=0}^{N-1}\frac{\zeta (2\cdot \frac{z+j+1}{2}-1)}{\zeta (2\cdot \frac{z+j+1}{2})}=\zeta (z)\wedge \underset{N\to \infty }{\lim }\prod _{j=0}^{N-1}\frac{\sqrt{\pi }\cdot \Gamma (\frac{z+j+1}{2}-\frac{1}{2})}{\Gamma (\frac{z+j+1}{2})}\cdot \frac{\zeta (2\cdot \frac{z+j+1}{2}-1)}{\zeta (2\cdot \frac{z+j+1}{2})}\cdot \frac{\Gamma (\frac{z+j+1}{2})}{\sqrt{\pi }\cdot \Gamma (\frac{z+j}{2})}=\zeta (z)$$

Proof. Machine-checked in Lean as D5/S3/Weil/Scattering/ScatteringZetaReconstruction.scattering_zeta_reconstruction (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every complex z with real part greater than one, the first displayed factor is the normalized zeta ratio at the shifted half-argument. Its finite products telescope exactly to zeta(z) divided by zeta(z+N).

A vertical-translate L-series with unit-modulus coefficients proves that zeta(z+N) tends to one for arbitrary fixed imaginary part. Gamma nonvanishing on the relevant right half-plane then cancels the Archimedean factors in the second displayed product.

References

• Truth anchor: D5/S3/Weil/Scattering/ScatteringZetaReconstruction.scattering_zeta_reconstruction