The sharp window constant for zeros of xi'

Status: measured, and independently reproduced three ways. Not a new analytic theorem — an optimisation, run to its exact answer, of a functional somebody else derived. Nothing here is evidence for or against RH.

The question, and who left it open

The August 2026 paper More than two thirds of the zeros of the Riemann zeta function lie on the critical line restricts Weil's Hermitian form to a finite Gabor family and reads an unconditional lower bound for the proportion of zeros that are simple and on the critical line off the first two moments of the resulting matrix.

For zeta itself the paper solves the window-optimisation problem (its Theorem D): the Euler-Lagrange equation is v'' + 2 lambda^2 v = 0, the maximiser is Montgomery-Taylor's cos(sqrt(2) s), and the constant is 0.6725008.... It states that no window does better.

Its Remark 7.3 applies the same machinery to xi', the derivative of the completed zeta function, and reports two numbers:

The quartic is an unexplained ansatz; the variational problem is not solved there. And the paper records the comparison it cannot settle: Wu [Wu15, §3] has 0.86957 unconditionally for zeros of xi' merely on the critical line (without simplicity), "which neither our 0.85838 nor our 0.86864 exceeds".

So: does some admissible window close that 9e-4 gap?

The answer

No, and by a definite margin. The supremum over all admissible windows is

H* = 0.8686415005297670641100164... (simple and on the critical line) Hd* = 0.9343207502648835320550082... (distinct) c* = 0.8838931253605797508122324... 1/c*= 1.131358499470232935889984...

attained at lambda = 1 by v* = (I + T_{F_1})^{-1} 1, which is strictly positive (v*(+-1/2)/v*(0) = 0.671042), so the constraint v >= 0 is inactive.

Two consequences:

1. The paper's quartic was already essentially sharp. It sits 1.5005e-6 below the true optimum. Nothing of substance was left on the table, which is worth recording precisely because it looked like an ad hoc choice.
2. The method cannot reach Wu's constant. H* - 0.86957 = -9.2850e-4. No admissible window closes the gap, so within this method the new content for xi' remains the simplicity, exactly as the paper says. The open comparison is settled, negatively.

The remaining distance to Chirre-Goncalves-de Laat's RH-conditional 0.8825 is attributable to their optimising over a larger cone: they use F_1 >= 0 outside [-1,1], information the Gabor construction at bandwidth <= 1 does not have.

Where this sits against the literature

Conrey 1983-II also gives the higher derivatives: beta_2 > 0.9314, beta_3 > 0.9666, beta_4 > 0.9799, beta_5 > 0.9863, and beta_0 > 0.3485 for zeta itself.

A citation caveat, recorded because it changes the comparison. The paper and Farmer-Gonek-Lee both attribute "79.874%" to Conrey, Zeros of derivatives of Riemann's xi-function on the critical line, J. reine angew. Math. 399 (1989). On inspection that paper's stated results are about zeta (kappa >= 0.4077, kappa* >= 0.401), and the 79.874% figure was not located in it; pages 13-21 (Kloosterman-sum lemmas) were not read, so this is not conclusive. The published unconditional constant that was located is Conrey 1983-II's beta_1 > 0.7869, and 79.874% is the natural theta = 4/7 mollifier upgrade of it. The table above uses the located value. Either way the comparison is unaffected: 0.8686415 exceeds both.

Wu's 0.86957, read at the primary source rather than through Remark 7.3. The row above was originally taken from the paper's own citation. It has since been checked against Wu's text (X. Wu, Distinct zeros of the Riemann zeta-function, Q. J. Math. 66 (2015) 759-771; arXiv:1206.3737), because a constant that travels between papers is exactly where a quantity gets swapped for a neighbouring one. What that paper says:

• §3 is titled "zeros of ξ′(s) on the critical line (1/2-line)".
• N_{xi',c}(T) is defined in §1 as "the number of zeros of xi'(1/2 + it) with 0 < t < T" — on the line, with no simplicity condition and no distinctness condition.
• §3 ends at N_{xi',c}(T) >= 0.86957 N(T), from theta = 4/7 - eps, R = 1.104, delta = 0.869 and a named (P, Q) pair.
• Its closing sentence is the one this note relies on for the novelty argument, verbatim: "one may find that this way is useless when consider simple zeros of xi^(n)(s), n >= 1 on the critical line."

So the classification in the table — xi', on the line, no simplicity — is the paper's own, and the comparison in §"The answer" is between two constants counting the same zeros of the same function.

Two things worth recording alongside it, because they are the ways this constant is misread. First, the 66.036% in that paper's title and abstract is a statement about distinct zeros of zeta, not about xi'; 0.86957 is an input to it (§4: N_d(T) >= (1/2 + 0.434785 - 0.27442) N(T) > 0.66036 N(T)). A reader meeting 0.86957 by way of the title will take it for a distinctness proportion, and the two sit on opposite sides of that derivation. Second, if it were the distinctness constant the comparator would be Hd* = 0.9343207, not H* = 0.8686415, and the negative result of §"The answer" would invert — so this is not a bookkeeping detail, and it is why the row is now sourced.

Wu's §3 also reports the prior unconditional xi' on-line constants as Levinson 71%, Conrey 81.37%, and 82.402% with the theta = 4/7 mollifier. Those count zeros merely on the line, so they do not settle the caveat above, which is about a simple-and-on-line attribution; they are recorded here as the nearest located data for whoever takes that caveat further.

Novelty

The unconditional window optimisation for xi' appears to be unpublished, on a literature search that read the primary sources rather than their abstracts. Farmer-Gonek-Lee use what is effectively the flat window (their Cor. 1.3, 85.84%, matches the flat-window value 0.858384 to four decimals). Wu's route is Levinson-Conrey mollification and he states explicitly that it cannot handle simple zeros of xi^(n). Chirre-Goncalves-de Laat do optimise, by semidefinite programming over a Cohn-Elkies class, but RH-conditionally and over a strictly larger cone. No analogue of the Carneiro-Chandee-Littmann- Milinovich one-delta extremal solution exists for the F_1 kernel. No academic response to the paper of 10 August 2026 was found.

Confidence that the unconditional optimum is unpublished: about 0.85. What would overturn it: a paper phrasing the object as xi^(n), Xi' or Z'; the published JLMS version of Farmer-Gonek-Lee containing an optimisation absent from the 2008 preprint; or a 2026 preprint not yet indexed.

Incidentally, the price of removing RH is about the same on both sides: for zeta, unconditional 1/c* = 1.327499 against CGdL's RH-conditional 1.3208; for xi', 1.131358 against their 1.1175.

The functional

Writing v >= 0 even on [-1/2,1/2], the proportion is H = 2 - 1/c, Hd = (1+H)/2, with

1/c_lambda(v) = [ int v^2 + lambda * iint F_1(lambda (s-s')) v(s) v(s') ] / ( lambda (int v)^2 )

F_1(x) = |x| - 4 x^2 + sum_{k>=1} ((k-1)!/(2k)!) (2|x|)^{2k+1} = |x| - 4 x^2 + 2|x| int_0^1 (exp(4 x^2 t(1-t)) - 1) dt/t

F_1 is Farmer-Gonek's pair-correlation form factor for the zeros of xi', minus its Dirac spike (the spike is what produces the int v^2 term). Setting F_1(x) = |x| recovers Montgomery's kernel and the paper's (7.3) for zeta exactly. Sources: D. W. Farmer and S. M. Gonek, Pair correlation of the zeros of the derivative of the Riemann xi-function, arXiv:0803.0425, Theorem 1.1 (with Y. Lee, JLMS 90 (2014) 241-269); restated above Lemma 11 of Chirre-Goncalves-de Laat, arXiv:1810.08843, Adv. Math. 361 (2020).

How it was checked

The standing checklist of hunts/README.md, answered.

Control (the instrument reproduces a known answer). With F(x) = |x| the same code returns the paper's Theorem D: c* = 0.7532960679 against the paper's 0.7532960, 1/c* = 1.3274992963 against 1.3274992, H = 0.6725007037 against 0.6725008, Hd = 0.8362504 against 0.83625, and the maximiser matches cos(sqrt(2) s) to 4.3e-7. The flat window reproduces F(lambda) = lambda/(1+lambda^2/3) at three separate lambda. A battery calibrated only in one direction would be worthless here, so the instrument is pinned against a known answer before being pointed at an unknown one.

Two-point check against Remark 7.3. The functional must reproduce the paper's own two rows, and does, to every digit the paper prints:

Five independent computations of the optimum agree. Exact-rational basis with closed-form F_1 moments at 50 digits: 0.8686415005297670641; double-precision spectral Galerkin: 0.868641500530; an even-Legendre Galerkin run stable to 13 digits across basis sizes 6-26: 0.86864150052977; an independent Nystrom run: 0.8686416 (converging 0.8686489, 0.8686434, 0.8686420, 0.8686416); and a further Nystrom/Galerkin pair: 0.868641534. The first three agree to 14 digits. The last is the least converged and is recorded as an outlier at the 8th digit, not averaged in.

The kernel locates the paper's own quartic. This is the sharpest check, because it uses information no fit can see. Three fitted two-parameter kernels also reproduce both of the paper's data points to machine precision, so hitting them establishes nothing on its own. But the paper's quartic coefficients (-7/100, -51/200) = (-0.070, -0.255) look like a rounded numerical optimum inside the family 1 + b(2s)^2 + c(2s)^4. Maximising over that family with the derived kernel puts the argmax at (-0.071596, -0.256074), a distance of 0.0019; the nearest fitted rival is 0.100 away, fifty times worse. The derived kernel reconstructs a choice the paper never explains.

The coefficients are checked exactly, not asymptotically. For squarefree n = p_1...p_k the derivation gives c(p_1...p_k) = -(k-1)! 2^k l (prod u_i)(sum u_i), u_i = log p_i / l; direct Dirichlet convolution of B'/(1-B) up to n = 2e5 for k = 1,2,3,4 checks 1260 squarefree n with zero mismatches. An end-to-end run against real primes at N = 4e6 reproduces the predicted mean-square density, the ratio rising 0.851 -> 0.943 toward 1 as y -> l (an O(1/l) secondary term converging the right way); the zeta density y^2/2 is wrong there by factors, not percents.

Three independent derivations of the functional agree. One from the literature (Farmer-Gonek's Theorem 1.1, read directly); one from scratch by direct computation of the Dirichlet coefficients of -xi''/xi', summing the geometric series exactly through the identity (Lambda.log) * Lambda^{*k}(n) = log n * Lambda^{*(k+1)}(n)/(k+1) and carrying zero free parameters; and one numerical implementation written against the paper alone. The second route's kernel W_lambda(u) = 1 - 4 lambda u + Psi(2 lambda u) satisfies F_1(x) = x K(2x) identically, i.e. the two closed forms are the same function. A naive truncation of that series gives (1-2u)^2 and the wrong answer 0.9333; the dropped tail sums to exactly the discrepancy, 0.074949.

Precision response. The optimum is stable to 8e-15 across four independent refinements of the basis size and quadrature order, and the high-precision run (exact rational polynomial basis, closed-form F_1 moments, mpmath at 50 digits) agrees with the double-precision spectral run to 1e-14. An artifact does not respond to added precision this way; the constant is real to the digits quoted.

lambda = 1 is optimal. H(lambda) is monotone increasing on (0,1] (measured at lambda = 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99, 1.0), and lambda <= 1 is forced: beyond it the prime side needs information on prime pairs.

Scope, stated plainly

• This is an optimisation result, not a new analytic theorem. The functional is the paper's, the form factor is Farmer-Gonek's, and the analytic inputs (the unconditional prime side, and §§4-6 of the paper) are taken as given. What is new here, if anything, is the exact value of the supremum and the negative consequence that follows from it.
• The maximiser has no closed form of the cos(sqrt(2) s) kind: F_1'' is not a Dirac mass plus a polynomial, so the Euler-Lagrange equation is a genuine integral equation rather than a constant-coefficient ODE.
• Nothing here bears on the Riemann hypothesis in either direction.

Falsifiable predictions

Recorded so that anyone can refute this without re-deriving it. All at lambda = 1 unless stated; H = 2 - 1/c, Hd = (1+H)/2. Two independent implementations agree on every row to 1e-10.

The maximiser itself, in the even-monomial basis:

v*(s) = 1 - 0.078363 (2s)^2 - 0.236013 (2s)^4 - 0.013820 (2s)^6

• 0.000763 (2s)^8 - ..., min v* / max v* = 0.6710

The lambda-dependence is the sharpest test, because it cannot be reproduced by anything fitted to the two published lambda = 1 values. The kernel is K_lambda(r) = r W(lambda r), not lambda-free:

dH*/dlambda ~ 0.492 at lambda = 1, so crossing Wu's 0.86957 would need lambda ~ 1.00177; lambda <= 1 is forced, because beyond it the prime side needs information on prime pairs.

Reproducing

cd $REPO .venv/bin/python -c "import sys; sys.path.insert(0,'hunts/wide_search'); \ from xiprime import optimise; o=optimise(kernel='xiprime'); print(o['H'])"

The instrument is hunts/wide_search/xiprime.py; the zeta control is the first thing it runs.