Higher derivatives: what blocks the extension, measured

Status: a measured obstruction and a located piece of buried prior art. Not a result about zeta. Nothing here is evidence for or against RH.

The obvious extension, and why it is not obvious

The paper's Remark 7.3 treats xi' only. Since its machinery turns a pair-correlation form factor into an unconditional proportion, the natural next move is xi^(k) for k >= 2, where Conrey's unconditional constants are higher and would have to be beaten.

Two things had to be established first: whether the form factor F_k is known, and whether it is usable.

The prior art, which is easy to miss

F_k for general k exists: Ji Bian, The Pair Correlation of Zeros of xi^(kappa)(s), PhD thesis, University of Rochester, 2008, supervised by Gonek — the companion to Farmer-Gonek-Lee. His Theorem 1 gives F_kappa(alpha) under RH for 0 < |alpha| < 1.

It was never published, is not on arXiv, is not in the arXiv full-text index, and carries essentially one citation in the literature. It is a genuine buried-prior-art hazard: a novelty claim for "the form factor of xi^(k)" made without it would have been wrong.

Two things it does not do, which is where the room is:

1. No closed form. Bian's coefficients come from an unevaluated ~14-fold combinatorial sum over set partitions and integral vectors (his eq. 7.8), evaluated by a Mathematica program reproduced in his Appendix A. There is no analogue of the k = 1 closed form 2|x| int_0^1 (exp(4 x^2 t(1-t)) - 1) dt/t.
2. No tail bound for kappa >= 2. In his words: "For higher cases of kappa, we can not prove that the tail of the function F_kappa(alpha) is small." His constants 0.9544 (k=2) and 0.9774 (k=3) rest on "assuming the coefficients after 11 terms are negligible", and he notes that for kappa = 3 "the coefficients are not settling down yet".

His Figure 10.1 tabulates exact rationals C_{kappa,i} in F_kappa(alpha) = spike + sum_i C_{kappa,i} |alpha|^i:

The kappa = 1 row is reproduced exactly — all eleven coefficients, exact rational equality — by the closed form used in RESULTS-xiprime.md. That is an independent check on the whole xi' computation from a source written sixteen years earlier, and it fixes the normalisation, so the rest of the table can be read with confidence.

The measured obstruction

Bian's caveat is not a formality. Feeding the 11-term truncations into the optimiser at lambda = 1:

For kappa = 4 the terms are still growing at i = 10 (2000.4). The truncation is usable only for alpha up to roughly 0.4, and the optimum sits at lambda = 1, which is exactly where it fails. kappa = 1 is the only row where truncating at eleven terms is harmless, and that is because its tail is known in closed form.

So no constant for kappa >= 2 is obtainable from the tabulated coefficients at the bandwidth the method needs. Bian's own numbers for k = 2, 3 are therefore RH plus an assumption that this measurement does not support at alpha near 1; whether they survive a real tail bound is open.

What would have to be true for the extension to work

A closed form for F_k, k >= 2, or a genuine bound on the tail. Either would also repair the gap Bian flagged himself. Two independent derivations were commissioned against the check grid above; their outcome is recorded separately if they reach one.

Worth knowing before anyone spends more on this: the headroom shrinks fast. Conrey's unconditional constants for simple-and-on-the-line are 0.79874 (k=1), 0.93469 (k=2), 0.9673 (k=3); the k=1 optimum reached here, 0.86864, recovers about 84% of the best RH-conditional constant. At the same efficiency k=2 would land near 0.951, about 1.6 points over Conrey, and k=3 near 0.976, under a point. Real, unconditional, and an order of magnitude less striking than k=1. (That extrapolation is an inference from read constants, not anything published.)

Provenance note

Conrey's unconditional constants are quoted from Conrey, Zeros of Derivatives of Riemann's Xi-Function on the Critical Line I and II (J. Number Theory 16 (1983) 49-74 and 17 (1983) 71-75), and from Farmer's 1995 restatement of Conrey's Crelle 399 (1989) results. The Crelle paper itself is paywalled and was not read; the alpha_j family for j >= 1 appears to be Farmer's combination of the 1989 and 1983-II results rather than a display in the 1989 paper, and is cited here as such. See RESULTS-xiprime.md for the related caveat about the 79.874% attribution.

TO BE CONTINUED

Two threads were live when this session ended and are not claimed. Both are picked up from here, not restarted.

1. A closed form for F_k, k >= 2 (or a genuine tail bound). Two independent derivations were mid-flight. One had reproduced Bian's k = 1 row exactly on all eleven coefficients, found its general-k formula failing the kappa >= 2 rows, and reported it had located the cause but had not yet landed a fix. The other was mid-way through a structural simplification. No general-k formula survived, so nothing from either is recorded as a result. The check grid they must satisfy is the kappa = 2,3,4 table above, plus Bian's stabilisation lemma. Do not fit to that grid: eleven targets per row makes fitting easy and worthless.

1. Whether zeta's own 0.6725 can be moved toward 0.68185. The paper's Remark 1.1 states an explicit ceiling of 0.68185 for any certificate reading bandwidth-one data configuration by configuration, while Theorem D attains 0.6725, so the truth for the best such certificate lies in [0.6725, 0.68185]. The well-posed question is whether the constraints from the whole family of admissible windows, which all describe the same zero configuration, beat the best single window; the paper's per-window certificate is what gives the lower end. This is the highest-value thread of the three and it was barely begun. A clean proof that the joint problem collapses to the single-window one would also be a real answer.