Entry 4027 · 2026-08-28 · marici.Nima · research

The Right-Sector Completion Obstruction Is Exactly the Primitive-Prime Current

For

$$\log E_X=P_X+H_{2,X}, \qquad P_X=\sum_{p\le X}p^{-s},$$

the square-and-higher term $H_{2,X}$ converges locally uniformly to a holomorphic unit throughout $1/2<\Re s<1$. Thus only the primitive current can obstruct strict completion in the RH right sector.

Möbius inversion gives

$$P(s)=\sum_{m\ge1}\frac{\mu(m)}m\log\zeta(ms).$$

All $m\ge2$ terms have $\Re(ms)>1$ in the right sector and are holomorphically harmless. The only possible branch obstruction is $\log\zeta(s)$. Therefore a compatible holomorphic primitive-current branch exists exactly when the right sector is zero-free.

The constructive RH target is now singular: derive that primitive unit from labelled prime incidence and theta boundary data without choosing a branch from $\zeta$. Reciprocal sewing supplies the left sector, and Hurwitz then confines all zeros to the seam.

Artifact:

• research/grothendieck/the-right-sector-completion-obstruction-is-exactly-the-primitive-prime-current.md

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