Golden Germ Product Abscissa

Abstract

The explicit golden exponent gives a prime-local Euler product whose exact absolute-convergence boundary is one over phi squared.

Theorem 1.1 (The golden germ product has its exact abscissa).

$$(\forall v\in \mathbb{N},o5Beta(v)=\sqrt{5}\times v+\frac{1}{\phi }-\mathrm{fract}((v+1)\times \phi ))\wedge o5Beta(1)={\phi }^2\wedge o5Beta(2)={\phi }^3\wedge (\forall sigma\in \mathbb{R},\mathrm{Summable}((p,v)\mapsto e^{-sigma\times \mathrm{goldenSpectrum}(p,v)})\iff \frac{1}{{\phi }^2}<sigma)\wedge (\forall s\in \mathbb{C},\frac{1}{{\phi }^2}<\Re (s)\Rightarrow \mathrm{HasProd}(p\mapsto \sum _{v\ge 0}{p}^{-s\times o5Beta(v)},\prod _{p\text{prime}}\sum _{v\ge 0}{p}^{-s\times o5Beta(v)})).$$

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

Source. Repository-derived.

Commentary.

The exponent is the canonical golden exponent already used by the Euler-germ family. Its first two positive values isolate the prime term and the faster tail.

The convergence equivalence includes divergence at the boundary. Above that boundary the canonical prime-local factors have the displayed infinite product.

References

• Truth anchor: D5/S3/Analytic/EulerGerm/GoldenGermProductAbscissa.golden_germ_product_abscissa