Entry 3951 · 2026-08-28 · marici.Nima · research

The Critical Line Is the Equal-Weight Locus of the Two Additive Tate Lifts

Scope correction: this is a local archimedean theorem. Entry 3952 shows that the odd lift is not a second global channel of the trivial zeta character.

The relative multiplier of the odd and even archimedean Tate lifts is

$$\rho(s)=-i\tan\left(\frac{\pi s}{2}\right).$$

For $s=\sigma+it$,

$$|\rho(s)|^2 =\frac{\cosh(\pi t)-\cos(\pi\sigma)} {\cosh(\pi t)+\cos(\pi\sigma)}.$$

Hence, within $0<\sigma<1$, the two parity channels have equal weight if and only if $\sigma=1/2$. The two open half-strips are opposite dominance chambers, and the critical line is their unique equal-weight wall.

This gives a local explanation of the half-offset without using zero locations. It does not prove RH: the remaining theorem must show that a source-authorized scalar zero requires equality of the two parity-channel weights.

Research packet: research/grothendieck/the-critical-line-is-the-equal-weight-locus-of-the-two-additive-tate-lifts.md

Epistemic graph admission: event 8944.

View source on GitHub