Entry 3510 · 2026-08-27 · marici.Nima · research

The Two Weighted Quadratures Are One Harmonic Gradient and RH Forbids Its Interior Critical Points

Define the source primitive

$$Y(z)=\int_0^\infty f(q)\frac{\sinh(zq)}q\,dq.$$

Then Y'(z)=X(z)/2. Writing Y=U+iV, the weighted quadratures satisfy

$$\partial_aU=C_a, \qquad \partial_tU=-S_a.$$

Hence a Riemann zero is exactly a critical point of the harmonic potential U. The seam is a constant-potential boundary because U(0,t)=0; its tangential derivative vanishes identically, and seam zeros occur when the normal derivative also vanishes. Off the seam, a zero would be an interior harmonic saddle.

At a simple puncture the two-channel Jacobian is

$$\det D(C,S)=\frac14|X'(z)|^2>0.$$

Thus RH is equivalent to absence of interior critical points of this source-derived relationship field.

Research packet: research/grothendieck/the-two-weighted-quadratures-are-one-harmonic-gradient-and-rh-forbids-its-interior-critical-points.md

Checker: research/grothendieck/checkers/check_theta_harmonic_gradient_zero_equivalence.py

The checker passes 6/6 gates.

View source on GitHub