Finite Weil Forms, the Prime Zeta Function and the Müntz Tail: Certified Bounds and Where the Route Stops

We study finite pieces of Weil’s quadratic form and the prime zeta function P(s) = Σp p−s: the forms Qa on L2(−a, a), localWeil matrices M0(Ω, L, N) built from the prime powers n ≤ eL, the factorisation eP = ζG, and the Müntz tails of log ζ and P. Results are labelled (A) computer-assisted proofs with fully enclosed errors, (B) floating point with a priori error budgets, and (C) numerical evidence. Proved, or computer-assisted (A). 1.49 · 10−38 ≤ λ∗(1.1) ≤ 3.60 · 10−38 for Q1.1, and for every a ≤ 1.1 the lowest eigenvalue is simple and isolated with an even eigenfunction; these are the hypotheses of the theorem of Connes and van Suijlekom, after an explicit identification with the form of Connes, Consani and Moscovici. A Perron–Frobenius argument gives them for a ≤ 0.3047 without computation. For the Davenport–Heilbronn function the first loss of Weil positivity lies in (1.1, 1.82]. A block inertia lemma organises the detectors of the local matrices; the threshold above which they are positive for every choice of the prime phases satisfies log(Ω∗/2π) ≍ eL/2 (with a sketched proof). For P: its continuation and singularities, eP = ζG with G zero-free in Re s > 1/2, an unconditional Müntz formula, the solutions of the boundary relation of the functional equation, the explicit continuation of H2 = P − L(s) + 1/2 L(2s), and the criteria of Volchkov and of Balazard–Saias–Yor for H2. The growth exponent of the Müntz tail of P at every point is exactly Θ − σ, with Θ the supremum of the real parts of the zeros. A mixed boundary problem in a strip, with Dirichlet data from the primes and Neumann data from the functional equation, gives a local certificate without bounds for S(t), run on windows of length 10 at heights up to 1015. Computations and evidence (B), (C). Certificates from the primes and unconditional inputs up to height 1815, and with signs of Z up to 10043.54; a local factorisation of ζ near a hypothetical zero, which recovers ζ′(ρ) from the primes and the other zeros; window tests for an extra zero; the Müntz tail against pair correlation and the Bohr–Jessen law; and the values of H2 in ball arithmetic. We also prove a strip flux criterion and a local form of Bagchi’s criterion for eP. The last section lists what is proved, what is computed and what is equivalent to RH, and collects the obstructions at large support: the margin of positivity decays doubly exponentially, the cost of a certificate grows doubly exponentially, the structural argument stops at a ≈ 0.3, and the Euler product has to enter below a ≈ 1.8. Nothing here proves the Riemann hypothesis. This version replaces the earlier versions of arXiv:1711.11416, whose argument for the convergence of the partial sums of P in the strip is withdrawn.