Airy derivative zeta values

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Numbers

$n$

$Z_N(n)$

1.371721164198448347271940235247035456480631873312936064234297483654710031530692292189237610296009733

comment: The inverse cubes sum to $1$. [1]

0.9408094761549732459334329572989795986174247654783995196827034516215991188266435002033194739951669816

0.9144807761322988981812934901646903043204212488752907094895316557698066876871281947928250735306731552

0.8952641349601161154707006147275282263800746461566653568912020237829293484827266591439061128403346477

0.8780888444113083628712040934790476253762631144465062183705232215134925109048672668564315090621558495

0.8616982576054679146457533945739719083083606369528300147617531245432582003791781109379197152141725848

0.8457455333807107671362693091016053002275181468165698568408709999680866467507005624012587639798301487

10:

0.8301272770778958145304681466985598411323024047214594861013811499385336226285970453527065070474857191

11:

0.8148092115511024771882476459715436653023411463899338743456583516822916006758871739429480329826569931

12:

0.7997773756583305201554128704482767711674323161593144902072199286440591287273761259769313379343861488

13:

0.7850239401242954053412561378332143474096795506731162812868050392724580976676300825232814680251127775

14:

0.7705429932310577241835470163838317596558022146595448201800484222936601930249080094487926136721888924

15:

0.7563292712978761357208372095049678782476305606161308093390718990067211795066732697883817953519970429

16:

0.7423777723070362216956726074900933524851263642278365710743614060704030004957052810227108210519953119

17:

0.7286836368997216375290295666408338009402380593194303769642835343557147782487357383816980195148636820

18:

0.7152421108121702163378912165074447706109903128878881977732359920938479484662674568422660927733420978

19:

0.7020485322268512344621308175034599648882671395223801137775682263739492716806855846435743677517783848

20:

0.6890983267788735186424350089559812869133350206790709758708000108142689189718615883689115309381734908

Definition

The Airy derivative zeta values are $Z_N(n)=\sum_{k=1}^{\infty}(-a_k^{\prime})^{-n}$ for integers $n\geq2$, where $0>a_1^{\prime}>a_2^{\prime}>\cdots$ are the zeros of $\operatorname{Ai}^{\prime}$. [1]

Parameters

$n$

— inverse-power order ($n\geq2$)

Formulas

(1)

Write $\kappa=-\operatorname{Ai}^{\prime}(0)/\operatorname{Ai}(0)=3^{1/3}\Gamma(2/3)/\Gamma(1/3)$, where $\Gamma$ is the Gamma function. [3]

(2)

$Z_N(2)=1/\kappa$ and $Z_N(3)=1$. [1] The recurrence (4) also gives $Z_N(4)=1/(2\kappa^2)$, $Z_N(5)=2/(3\kappa)$ and $Z_N(6)=(1+\kappa^{-3})/4$.

(3)

For $|z|<-a_1^{\prime}$, the logarithm analytic near $z=0$ and equal to zero there satisfies $\log\!\left(\operatorname{Ai}^{\prime}(z)/\operatorname{Ai}^{\prime}(0)\right)=\sum_{n=2}^{\infty}\frac{(-1)^{n-1}}{n}Z_N(n)z^n$. [1]

(4)

For integers $n\geq4$, $(n-2)Z_N(n)=\sum_{j=2}^{n-2}Z_N(j)Z_N(n-j)$, with initial values $Z_N(2)=1/\kappa$ and $Z_N(3)=1$. This follows from $h^{\prime}(z)=1-zh(z)^2$ for $h(z)=\operatorname{Ai}(z)/\operatorname{Ai}^{\prime}(z)$, using the Airy equation $\operatorname{Ai}^{\prime\prime}(z)=z\operatorname{Ai}(z)$. [3]

Comments

(5)

The positive numbers $-a_k^{\prime}$ are the eigenvalues of $H=-d^2/dx^2+x$ on $x\geq0$, with the Neumann condition $u^{\prime}(0)=0$ and decay at infinity. Each eigenvalue is counted once. The kinetic coefficient and potential slope are both $1$, and the additive energy shift is $0$. Thus $Z_N(n)=\operatorname{Tr}(H^{-n})$. This is the even-parity sector $Z^+(n)$ in [1]. The zeros $a_k^{\prime}$ are the locations of the extrema of $\operatorname{Ai}$.

(6)

The zero asymptotic $-a_k^{\prime}\sim[\frac{3\pi}{2}(k-\frac34)]^{2/3}$ implies convergence for real order greater than $3/2$. [2] All entries are convergent sums; no analytic continuation is needed.

(7)

The negative-root convention gives $\sum_{k\geq1}(a_k^{\prime})^{-n}=(-1)^n Z_N(n)$. The entries use the positive spectrum, so every value is positive.

(8)

The Airy zeta values use the Dirichlet condition $u(0)=0$ and the zeros of $\operatorname{Ai}$ in place of the Neumann condition and the zeros of $\operatorname{Ai}^{\prime}$.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

B = RealBallField(429)
t = B(QQ(1)/3).gamma() / (B(3)**(QQ(1)/3) * B(QQ(2)/3).gamma())
Z = {2: t, 3: QQ(1)}
for n in range(4, 21):
    Z[n] = sum((Z[j]*Z[n-j] for j in range(2, n-1)), B(0)) / QQ(n-2)
for n in range(2, 21):
    print(n, Z[n])

References

[1]

A. Voros, Airy function (exact WKB results for potentials of odd degree), Journal of Physics A 32 (1999), 1301–1311, Section 3, equations (28), (29) and (43.2)–(43.3). (arXiv) (doi)

Links

[2]

NIST DLMF 9.9: Airy zeros and their asymptotics

[3]

NIST DLMF 9.2: Airy equation and initial values

Similar tables

Local extrema of the Airy function $\operatorname{Ai}$ — The negative derivative zeros whose absolute values form the Neumann spectrum.

Values of $\operatorname{Ai}^{\prime}$ — Values of the function whose logarithmic expansion yields these spectral moments.

Airy zeta values — The inverse spectral moments for the Dirichlet boundary condition of the same half-line operator.

Data properties

Entries are of type: real number

Table is complete: no (it holds every integer order $2\leq n\leq20$, an initial segment of the convergent inverse-operator moments and logarithmic spectral-determinant coefficients; the upper bound is an editorial cutoff for this discrete family)

How well the digits are known: proven

How they were obtained:

Each value was obtained by exact rational-polynomial arithmetic followed by evaluation in Arb balls. The Gamma arguments and cube-root exponent were exact rationals. A 96-bit guard gives 429 working bits for 100 significant digits; the lowest measured relative accuracy was 421 bits and the largest relative radius was less than 7.4e-128. Every value was recomputed at 628 bits.

An independent 600-bit computation extracted all 19 coefficients of Ai(z)/Ai′(z) from 512 direct Airy-function evaluations on the circle |z|=1/2, with an explicit alias bound of 2/(2^512-1). A polynomial with known nonzero coefficients was tested first as a control. The published low-order identities, the recurrence, positivity, strict decrease and strict log convexity were also checked. The exact identity Z_N(3)=1 is preserved as an integer. The attached checks verify containment of the independent enclosures in the stored decimal intervals and rerun the identities on values read back from the database.