Golden Germ Zeta Residue

Abstract

The simple golden boundary pole has the explicit positive residue G(a) over phi squared.

Theorem 1.1 (The golden boundary residue is explicit and positive).

$$\begin{array}{r}G:\mathbb{C}\to \mathbb{C},\\ \forall s\in \mathbb{C},G(s):=\prod _{p\in \mathrm{Primes}(\mathbb{N})}(1-p^{-s\times {\phi }^2})\times \sum _{v\in \mathbb{N}}{p}^{-s\times o5Beta(v)},\\ Z:\mathbb{C}\to \mathbb{C},\\ \forall s\in \mathbb{C},Z(s):=\mathrm{riemannZeta}({\phi }^2\times s)\times G(s),\\ a:=\frac{1}{{\phi }^2},\\ \mathrm{meromorphicOrderAt}(Z,\frac{1}{{\phi }^2})=-1\wedge \\ \mathrm{Tendsto}((s:\mathbb{C})\mapsto (s-\frac{1}{{\phi }^2})\times Z(s),\mathrm{nhdsWithin}(\frac{1}{{\phi }^2},\mathbb{C}\setminus \{\frac{1}{{\phi }^2}\}),\mathrm{nhds}(G(\frac{1}{{\phi }^2})/{\phi }^2))\wedge \\ \mathrm{Im}(G(\frac{1}{{\phi }^2})/{\phi }^2)=0\wedge 0<\Re (G(\frac{1}{{\phi }^2})/{\phi }^2).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Analytic/Isolation/GoldenGermZetaResidue.golden_germ_zeta_residue (✓ std3). ∎

Source. Repository-derived.

Commentary.

This theorem is the next boundary node in the golden Euler germ extraction ladder of OACTC parts 580 and 581, on the RH-route O-5 control line. The frozen predecessor identifies a genuine simple pole at a equal to one over phi squared; this node closes the remaining explicit-residue boundary by computing its value.

GoldenGermZetaBoundary supplies the transported limit of the zeta kernel and the exact factorization of (s-a)Z(s). GoldenGermNormalizedFactorRegularity makes G continuous at a, so the product limit is G(a) over phi squared. Frozen real-axis positivity of G(a), together with positivity of phi squared, makes this residue real and strictly positive.

GoldenGermZetaSimplePole supplies the meromorphic order minus one. STOPPING JUSTIFICATION: the conclusion concerns only the point one over phi squared and its displayed punctured neighborhood. It does not assert O-5, the Riemann hypothesis, any implication toward either claim, a zero-free region, or a pole at any other point.

References

• Truth anchor: D5/S3/Analytic/Isolation/GoldenGermZetaResidue.golden_germ_zeta_residue
• Dependency: D5/S3/Analytic/EulerGerm/GoldenGermZetaBoundary
• Dependency: D5/S3/Analytic/Isolation/GoldenGermZetaSimplePole
• Dependency: D5/S3/Analytic/Regularity/GoldenGermNormalizedFactorRegularity