Riemann Poisson Density

Abstract

The shifted-xi phase density is the Poisson smoothing of its zero-counting measure.

Theorem 1.1 (Riemann Poisson-density theorem).

$$\forall Z\in \mathrm{ZeroData}\left(\right),omega\in \mathbb{R},\left(0<omega\wedge \left(\mathrm{RiemannHypothesis}\left(\right)\wedge \left(\forall x\in \mathbb{R},0<omega\Rightarrow \left(\mathrm{RiemannHypothesis}\left(\right)\Rightarrow \mathrm{phaseDensity}\left(omega,x\right)=\sum _{n\in \mathbb{N}}\mathrm{multiplicity}\left(Z,n\right)\cdot \mathrm{poissonKernel}\left(omega,x-\mathrm{im}\left(\mathrm{zero}\left(Z,n\right)\right)\right)\right)\right)\right)\right)\Rightarrow (x\mapsto \mathrm{phaseDensity}\left(omega,x\right))=(x\mapsto \mathrm{poissonSmooth}\left(omega,\mathrm{zeroCountingMeasure}\left(Z\right),x\right))$$

Proof. Machine-checked in Lean as D5/S3/Weil/ZetaAnalytic/RiemannPoissonDensity.riemann_poisson_density (✓ std3). ∎

Source. Repository-derived.

Commentary.

The phase density is constructed from the logarithmic derivative of the canonical entire xi reading. The counting measure is built independently from a duplicate-free exhaustive zero enumeration.

Under the critical-line hypothesis and the preceding logarithmic-derivative zero expansion, integration against the weighted sum of Dirac masses is exactly the sum of translated Poisson kernels.

References

• Truth anchor: D5/S3/Weil/ZetaAnalytic/RiemannPoissonDensity.riemann_poisson_density
• Dependency: D5/S3/Zeros/CompletedZeta