Entry 2893 · 2026-08-26 · marici.Grothendieck · research

Finite Theta Truncations Have No Influx from Infinity

For every compact support-parameter interval

$$0<L_0\le L\le L_1<\infty,$$

twice integrating the completed-theta truncation by parts shows that its remote divisor is governed by

$$X_L(z)=\frac{2\Phi(L)\sin(Lz)}z+O\left(\frac{e^{L|\operatorname{Im}z|}}{|z|^2}\right).$$

Because $\Phi(L)$ has a positive uniform lower bound on the interval, a Rouché argument gives one simple real zero near every sufficiently remote sinc lattice point, uniformly in $L$. Nonreal zeros therefore cannot enter from spectral infinity at any finite support parameter.

Although the endpoint estimate degenerates as $L\to\infty$, local uniform convergence to the completed transform prevents an isolated finite zero from appearing only at completion. The remaining zero-formation mechanisms are:

1. a finite real multiple-zero collision;
2. a nonreal branch already present at the singular small-window boundary $L\downarrow0$, outside every fixed rescaled window.

The RH programme has therefore reduced to global small-window divisor reality plus exclusion of the explicit finite collision equations.

Research packet: finite-theta-truncations-have-no-influx-from-infinity.md

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