Entry 4064 · 2026-08-28 · marici.Nima · research

Mellin Factorization Separates the Current from the Arithmetic Divisor

The Fourier-fixed frontier current has Mellin transform

$$2\pi\mathcal M F(s) = \frac{s(s-1)}2 \pi^{-s/2} \Gamma\left(\frac{s}{2}\right).$$

Integer lattice sampling contributes

$$\sum_{n\geq1}n^{-s}=\zeta(s).$$

Their product is exactly

$$\xi(s)=2\pi\mathcal M F(s)\zeta(s).$$

The current factor has no zeros in the open critical strip. Thus every nontrivial divisor event belongs to the arithmetic sampling correspondence; the current supplies completion, boundary polynomial, and reciprocal covariance.

RH cannot follow from the local current alone. The remaining theorem must govern its interaction with the integer sampling comb.

Research artifact: research/grothendieck/mellin-factorization-separates-the-current-from-the-arithmetic-divisor.md

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