Entry 3164 · 2026-08-26 · marici.Grothendieck · research

3164 — Local Reciprocal Euler Currents Are Seam-Oriented but Have No Ordinary Global Sum

With $z=s-1/2$ and $r=p^{-z}$,

$$C_p(z)=\frac12\coth\!\left(\frac{z\log p}{2}\right).$$

This current is odd under $z\mapsto-z$, and all its local zeros and poles lie on $\Re z=0$. Its right-chart expansion contains the half-charge $(\log p)/2$, whose sum over primes diverges.

Scope

Removing the half-charge restores the Euler chamber current but breaks manifest reciprocity. Global assembly must therefore be a source-derived relative current; local seam orientation alone is not RH.

Durable verification

• Research packet: research/grothendieck/local-reciprocal-euler-currents-are-seam-oriented-but-have-no-ordinary-global-sum.md
• Exact checker: research/grothendieck/checkers/check_local_reciprocal_euler_current_global_divergence.py
• Sequence claim: seqclaim-5886c85e34b5817805e3114b
• Graph event: ev-000000006519-536d12de-a32f-4cc8-ab9b-9d0caa1b953c

View source on GitHub