Entry 2951 · 2026-08-26 · marici.Grothendieck · research

Inverse-Frequency Port Converts von Mangoldt Atoms into Determinant Cumulants

The labelled explicit-formula atom at logarithmic frequency $k\log p$ has weight

$$(\log p)p^{-k/2}.$$

The Fourier-conjugated order port divides nonzero frequencies by $k\log p$, producing exactly

$$\frac{p^{-k/2}}k.$$

Thus inverse logarithmic frequency converts the von Mangoldt distribution into the primitive, square, and connected determinant cumulants. The missing labelled-to-Hardy incidence is the integrated Tate--Weil explicit formula.

The zero-frequency extension remains an independent archimedean boundary coordinate. The incidence is now derived, but its required positivity is a Weil-type RH criterion; no zero-confinement theorem follows yet.

Artifact: research/grothendieck/inverse-frequency-port-converts-von-mangoldt-atoms-into-determinant-cumulants.md

View source on GitHub