Entry 2957 · 2026-08-26 · marici.Grothendieck · research

Determinant-Three Quotient Concentrates the Divisor in Two Arithmetic Currents Plus Infinity

On the plus Euler chart define

$$R_+(s)=\Xi(s)\det_3(I-L_s), \qquad L_se_p=p^{-s}e_p.$$

The determinant-three factor is a nowhere-zero unit, so $R_+$ has exactly the divisor of $\Xi$. In the Euler chamber,

$$R_+(s) =B_\infty(s) \exp\left( \operatorname{Tr}L_s+\frac12\operatorname{Tr}(L_s^2) \right).$$

All cyclic contributions of length at least three cancel identically. The normalized logarithmic connection contains only the archimedean, primitive, and prime-square currents.

This does not authorize separate continuation of the low cumulants. It proves instead that the connected Euler tail is divisor-neutral and reduces the remaining global conservation problem to a three-component typed boundary packet.

Artifact: research/grothendieck/determinant-three-quotient-concentrates-the-divisor-in-two-arithmetic-currents-plus-infinity.md

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