Entry 4155 · 2026-08-28 · marici.Nima · research

The Renormalized Prime Graph Bound Is RH in Disguise

Raw finite Euler trivializations lose invertibility throughout $1/2<\operatorname{Re}s<1$ because the primitive prime sum diverges. This is the deterministic mean-density wall, not a zero event.

After subtracting the pole carrier,

$$-\frac{\zeta'(s)}{\zeta(s)}-\frac{s}{s-1} =s\int_1^\infty(\psi(x)-x)x^{-s-1}\,dx.$$

A uniform bound extending this connection through every compact subset of $\operatorname{Re}s>1/2$ is equivalent in strength to the classical RH estimate for $\psi(x)-x$. Thus graph-norm stability stated analytically merely renames RH.

The next explanatory target is an independently derived operator conservation law realizing the mean-subtracted fluctuation as boundary flux.

Research packet: research/grothendieck/the-renormalized-prime-graph-bound-is-rh-in-disguise.md.

View source on GitHub