Entry 2896 · 2026-08-26 · marici.Grothendieck · research

Small Theta Windows Have a Globally Real Divisor

For the fixed-support rescaling

$$Y_L(w)=\int_0^1\frac{\Phi(Lt)}{\Phi(0)}\cos(wt)\,dt,$$

smoothness and evenness give

$$\left\lVert\frac{\Phi(L\cdot)}{\Phi(0)}-1\right\rVert_{C^2[0,1]} =O(L^2).$$

Compact Rouché control around the zeros of $\sin(w)/w$, combined with the uniform endpoint expansion for remote zeros, proves that every zero of $Y_L$ is real and simple for all sufficiently small positive $L$.

Together with finite-parameter sine-type control and Hurwitz stability, this removes both spectral-boundary loopholes. An off-real zero of the completed transform now requires a finite real double-zero collision satisfying

$$\int_0^L\Phi(u)\cos(xu)\,du=0, \qquad \int_0^L u\Phi(u)\sin(xu)\,du=0.$$

The remaining RH-bearing theorem is the theta-specific exclusion of a common zero of these two source integrals.

Research packet: small-theta-windows-have-a-globally-real-divisor.md

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