Pole-Continuum Completion

Abstract

The completed-zeta pole pair minus the continuous prime density is the decaying Green-kernel form, and its multiplier advances the digamma argument by one.

Theorem 1.1 (The pole-continuum difference is the decaying Green form).

$$\forall f\in \mathcal{W},\mathrm{poleTerm}\left(\mathrm{convolutionSquare}\left(f\right)\right)-{\int }_0^{\infty }\exp \left(\frac{u}{2}\right)(\mathrm{convolutionSquare}\left(f,u\right)+\mathrm{convolutionSquare}\left(f,-u\right))du={\int }_{\mathbb{R}}{\int }_{\mathbb{R}}\exp \left(-\frac{\left|x-y\right|}{2}\right)f\left(x\right)\overline{f\left(y\right)}dydx$$

Proof. Machine-checked in Lean as D5/S3/Weil/ZetaGamma/PoleContinuumCompletion.pole_continuum_completion (✓ std3). ∎

Source. Repository-derived.

Commentary.

W is the canonical carrier of even smooth compactly supported complex tests. The displayed half-line integral is the continuous prime main density evaluated on the canonical convolution square.

Splitting the two pole evaluations into growing and decaying exponentials cancels the growing half-line contribution. Fubini and translation invariance identify the remainder with the displayed full-line Green kernel.

Theorem 1.2 (The Green multiplier advances the digamma argument by one).

$$\begin{array}{r}\forall xi\in \mathbb{R},\\ \mathrm{let}{b}_{\infty }=\Re (\psi (\frac{1}{4}+\frac{ixi}{2}))-\log \left(\pi \right)+\frac{1}{x{i}^2+\frac{1}{4}},\\ b_{\infty }=\Re (\psi (\frac{5}{4}+\frac{ixi}{2}))-\log \left(\pi \right)\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Weil/ZetaGamma/PoleContinuumCompletion.archimedean_shift_completion (✓ std3). ∎

Source. Repository-derived.

Commentary.

The identity is the exact digamma recurrence at one quarter plus half the imaginary frequency. Taking real parts turns the reciprocal term into the displayed Green multiplier.

References

• Truth anchor: D5/S3/Weil/ZetaGamma/PoleContinuumCompletion.archimedean_shift_completion
• Truth anchor: D5/S3/Weil/ZetaGamma/PoleContinuumCompletion.pole_continuum_completion