Opened 2026-09-10. Nothing in this directory is a result. It is a hunt (hunts/README.md): exploratory work, recorded before any control has been run against it, and the strongest words it may use are measured, observed and consistent with. The reserved enclosure word belongs to zeta/rigor.py and appears nowhere here except where a quoted artifact forced a visible mask, which RESULTS.md records.
Scope, stated once: nothing in this hunt is evidence about the Riemann Hypothesis (docs/08). The Davenport-Heilbronn function is not zeta, it has no Euler product, and it already violates its own analogue of RH. Every number below is about that function.
The question
Bombieri and Lagarias (1999, Theorem 1) prove an equivalence for any multiset R of complex numbers closed under rho -> 1 - rho and satisfying their convergence condition:
lambda_n(R) = sum_rho [ 1 - (1 - 1/rho)^n ] >= 0 for every n >= 1 <=> every rho in R has Re rho = 1/2.
The Davenport-Heilbronn function f has an off-line zero pair measured at 0.80851718... + 85.69934848...i (zeta.epstein.OFFLINE_ZERO_RE, hunts/flow_repair), so the equivalence says that some lambda_n(DH) is negative. No one appears to have computed the sequence.
What are the Li coefficients of the Davenport-Heilbronn function, how far can n be pushed unconditionally on one 4-vCPU container, are they all positive there, and where does a DH background asymptotic measured from those same coefficients put the first negative index?
The route is fixed in advance and it is the one that never reads a zero list: mirror zeta.li._li_cauchy for the completed function
F(s) = (pi/5)^{-(s+1)/2} Gamma((s+1)/2) f(s), F(s) = F(1-s),
substitute s = 1/(1-z), unwrap the logarithm along the sampling circle, check the winding number, and read lambda_n = n * [z^n] log F(1/(1-z)) off one equispaced trapezoid DFT. The analyticity radius is established first, by the argument principle, and not assumed.
Why the radius argument is the whole gate
For any zero rho, z = 1 - 1/rho has |z| = |rho - 1|/|rho|, so |z| <= r holds exactly on the Apollonius disc |s - 1/(1-r^2)| <= r/(1-r^2). Equivalently every zero with |rho| >= 1/(1-r) is outside the disc of radius r. Zeros on the critical line map to |z| = 1 exactly and can never enter. So the radius is governed by low-lying zeros with Re > 1/2 only, and r = 0.9 needs f to have no zero inside the disc centred 5.263157... of radius 4.736842..., whose part with Re s <= 2 is contained in the rectangle [0.526, 2] x [-3.44, 3.44]. Above Re s = 2 zeta.epstein already pins sum_{n>=2} |a_n| n^{-2} = 0.2666... < 1, so f cannot vanish there. This is the same shape of argument as zeta.li.RADIUS_MAX, and like it, it appeals to no unproved hypothesis.
Preregistered predictions
- P1. The identical pipeline pointed at
zeta.core.xireproduceszeta.li.li_coefficientsto the digits that module states. Run first. - P2. lambda_n(DH) extracted at r in {0.5, 0.7, 0.9} agree to within the round-off amplification r^{-n} that the working-precision rule already charges for. Disagreement beyond it means the branch or the winding check is wrong.
- P3. The argument-principle scan finds no zero of f with Re s > 1/2 and |rho| < 10, so r = 0.9 is admissible.
- P4. Every computed lambda_n(DH) with n <= n_max is positive, because the dominant off-line quadruple's amplitude 2(R^n + R^-n) with R - 1 = 4.2e-5 is still O(1) at n = 2000 while the background is O(n log n).
- P5. A background of the form a n log n + b n + c fitted to the measured lambda_n(DH) recovers a = 1/2 and b = (log(5/(2 pi)) + euler - 1)/2 = -0.3256..., the conductor-5 analogue of
zeta.li.li_asymptotic, to within the residual oscillation. - P6. The first negative index implied by that measured background plus the fifteen known off-line quadruples lands in [1e5, 1e6].
What this hunt is not allowed to do
It may not touch zeta/, tests/, docs/, hunts/README.md or any other hunt. It may not run git. It may not call a positive lambda_n table evidence that the Davenport-Heilbronn function satisfies anything, and it may not call a computed sequence a theorem.
id: li_dh_onset
question: What are the Li coefficients lambda_n of the Davenport-Heilbronn function, how far can n be pushed unconditionally on one container, are they all positive there, and where does a measured DH background put the first negative index?
frontier: no published lambda_n(DH) at any n; the one prior estimate is hunts/jensen_clock phase 3 Q6, n ~ 3.3e5, from a zeta-shaped background and hedged by its own page as order-of-magnitude
proposed_attack: mirror zeta.li._li_cauchy on the completed DH function, equispaced trapezoid DFT of the unwrapped log of F(1/(1-z)) on |z| = r, with the analyticity radius established first by the argument principle
dead_routes:
- the envelope-crossing indicator of hunts/jensen_clock phase 3, which compared 2 r^n against lambda_n itself and fired at n = 1 because lambda_1 is 0.023; that page declares it useless and it is not reused here
- reading lambda_n(DH) off a zero list as the primary route, which would import every completeness assumption the zero hunt carries
required_oracles:
- the same Cauchy pipeline pointed at xi, reproducing zeta.li.li_coefficients to its stated digits
- radius independence of the extracted coefficients across r in 0.5, 0.7, 0.9
- zeta.epstein.dh_functional_equation_defect and dh_mean_value_defect on the evaluator
- argument-principle zero counts from zeta.epstein.count_zeros_box
- a partial zero-side reconstruction from measured on-line ordinates and the known off-line quadruples
kill_conditions:
- the extracted lambda_n(DH) differ across radii beyond the stated round-off amplification, in which case the branch or the winding check is wrong and every number here is withdrawn
- the xi control fails to reproduce zeta.li.li_coefficients
- the argument-principle scan finds a zero of f with Re s > 1/2 and modulus small enough to force the radius below 0.7, in which case only the small-n table survives
- a computed lambda_n(DH) goes negative at small n, which is first a defect in this pipeline and not a discovery
agents_may:
- derive
- code
- measure
- attack their own pipeline
- state a grade on the certainty ladder
agents_may_not:
- declare novelty
- declare theorem status
- promote their own claim
- write any sentence implying a computation bears on the Riemann Hypothesis