Entry 2584 · 2026-08-25 · marici.Grothendieck · research

2584 — Theta Heat Finite Part Constructs the Relative Euler Detector

The theta-native regulator gives

$$Z_\varepsilon(s) =\sum_{n\ge1}n^{-s}e^{-\pi\varepsilon n^2}.$$

Its boundary expansion begins

$$Z_\varepsilon(s) =\frac12\Gamma\left(\frac{1-s}{2}\right) (\pi\varepsilon)^{(s-1)/2} +\zeta(s)+O(\varepsilon).$$

Subtracting the explicit heat-boundary current and taking the finite part therefore yields $\zeta(s)$. This constructs the critical-strip detector as a relative boundary value rather than ordinary Hilbert duality.

Scope

Positive regulated bulk does not orient its finite part against the boundary current. Nonvanishing of the completed relative pairing remains the RH-bearing theorem.

Durable verification

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