Entry 4030 · 2026-08-28 · marici.Nima · research

RH Is Continuity of Connected Realization, Not Existence of the Euler Logarithm

The formal Euler object always has a plethystic logarithm:

$$H\log Z = \sum_{p,k\ge1}(\log p)X_p^k.$$

Realizing $X_p^k$ at the logarithmic atom $k\log p$, subtracting the continuum current $e^u du$, and twisting by $e^{-u/2}$ gives

$$\tau=e^{-u/2} \left( \sum_n\Lambda(n)\delta_{\log n}-e^u du \right).$$

For the cumulative discrepancy $q(u)=e^{-u/2}(\psi(e^u)-e^u)$,

$$(\partial_u+1/2)q=\tau$$

on the positive ray, with an explicit origin delta after extension by zero. Hence $q$ is tempered exactly when this connected boundary realization is tempered.

The missing RH theorem is therefore continuity of formal connectedization through completed theta realization into the half-normalized tempered rigging. Formal existence of the Euler logarithm is automatic and has no zero-selection force.

Artifact:

• research/grothendieck/rh-is-continuity-of-connected-realization-not-existence-of-the-euler-logarithm.md

View source on GitHub