Halves of the Riemann zeta function II: alignment, genealogy, and where the arithmetic enters

A half of the Riemann zeta function (or of another L-function S) is a function F with S(s) = F(s) + χ(s)F*(1−s). Part I proved that the number of off-line pairs of zeros of S equals Z⁺_F − R_F: the number of zeros of F to the right of the critical line minus the number of backward crossings of its phase. This paper continues in five directions.

(1) For polynomial halves the identity is an inertia statement. Restricted to the zeros of S, the Schur–Cohn form of F splits into one 1×1 block per zero on the line and one hyperbolic 2×2 block per off-line pair.

(2) The local model of a single well is solved exactly. In the model, a backward crossing is captured if and only if an alignment invariant A = S(ρ′)/(2d·F′(ρ′)) lies in an explicit region Ω. The δ/π law observed in Part I for random halves follows.

(3) Varying the length of the Riemann–Siegel main sum conserves Z⁺ − R. Each right zero of the half is either born on the line or inherited from an off-line zero. Exact computations with the Davenport–Heilbronn function, which has a functional equation but no Euler product, show that the counting identity and the geometry of wells are the same as for L(s, χ₋₄). The arithmetic enters only through the alignment.

(4) Levinson's half has no backward crossings, and the zeros of W = ζ + ζ′/L̃ are exactly the mirror images of the zeros of ζ′. So the mollifier-independent part of Levinson's loss is governed by the zeros of ζ′ near the critical line. A suitable half with no backward crossings gives an exact, windowed form of the Levinson–Montgomery theorem: in every tame window, the number of zeros of ζ′ to the left of the critical line equals the number of zeros of ζ to its right. This is verified on the Davenport–Heilbronn function. The constant term of the Eisenstein series is the case Z⁺ = 0 (Lagarias–Suzuki). At Heegner points of class number one, the author's uniform zero-free region gives GRH for Q(√−163) up to height 31.4.

(5) An exact sum rule for the alignment invariants and an explicit formula for the right zeros of a half show that the arithmetic coefficient Λ_S(x) appears only on the right edge of the contour. Moving it onto the zeros requires a mean value on the critical line at scale 1/log t, the same obstruction met by the other routes.

Nothing here proves the Riemann hypothesis. The paper states what is proved, what is observed numerically, and where each argument stops.