Golden Germ Zeta Factorization

Abstract

The golden germ product factors through the Riemann zeta function, with an absolutely convergent normalized product that is positive on its real ray.

Theorem 1.1 (The golden germ has a positive zeta-normalized factor).

$$G(s):=\prod _{p\text{prime}}(1-p^{-s\times {\phi }^2})\times \sum _{v\ge 0}{p}^{-s\times o5Beta(v)},(\forall s\in \mathbb{C},\frac{1}{{\phi }^2}<\Re (s)\Rightarrow \prod _{p\text{prime}}\sum _{v\ge 0}{p}^{-s\times o5Beta(v)}=\mathrm{riemannZeta}({\phi }^2\times s)\times G(s))\wedge (\forall s\in \mathbb{C},\frac{1}{{\phi }^3}<\Re (s)\Rightarrow \mathrm{Summable}(p\mapsto |(1-p^{-s\times {\phi }^2})\times \sum _{v\ge 0}{p}^{-s\times o5Beta(v)}-1|))\wedge (\forall sigma\in \mathbb{R},\frac{1}{{\phi }^3}<sigma\Rightarrow 0<\Re (G(sigma))\wedge \mathrm{Im}(G(sigma))=0).$$

Proof. Machine-checked in Lean as D5/S3/Analytic/EulerGerm/GoldenGermZetaFactorization.golden_germ_zeta_factorization (✓ std3). ∎

Source. Repository-derived.

Commentary.

The normalized factor is the source-defined Euler product itself: each prime-local golden germ series is multiplied by the inverse of its first zeta mode.

First-order cancellation leaves the beta-two tail and the square of the beta-one mode. Their prime sums converge above one over phi cubed, which proves absolute convergence of the displayed product.

On the real ray every local series and every cancelling factor is positive. The real infinite product is nonzero by summable deviations, and its complex embedding has positive real part and zero imaginary part. The source’s numerical window certificate is an empirical remark outside the named theorem.

References

• Truth anchor: D5/S3/Analytic/EulerGerm/GoldenGermZetaFactorization.golden_germ_zeta_factorization