Airy zeta values

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Numbers

$n$

$Z_D(n)$

0.5314572319609994528671249327835975128793762613598310302129500019359222956528578262843381519744913357

0.1125617612151145794304351152459810519503961166363151095605355424412345609571039396593371385193231989

0.03944307842123858454386231559273186269551737139039066301468634771716695964798177446256180160638766378

0.01553365937662315960105326698601338207533195387154164863322026920862587812029184977321978498881146467

0.006389269480291183085985031797193603176325864055526664641413635547275766261945530323301383069312594316

0.002679136468944483094339090139571816658235339260887016458460696566390563589062212139425800604794000764

0.001133964981406675717919109806835643688471429708423116819424855547821787518793036055882728723770737149

0.0004822640316384375106012087634290054281974234596780313421396461581118731301564303715306774345263986554

10:

0.0002056235790077167372193931809423657299790322916072759439435826244283175419560451917355215230540860066

11:

0.00008779281002346230360918430493634776036848020023646123963358579383445225392587069304424880757488041385

12:

0.00003751235204799617036542815440106965073975378474704740995369017970891205439852824635387534846506800370

13:

0.00001603514706599931127290224527419597145818156452369145789514814934905293590648796949856827081540509454

14:

0.000006856056842867739305355803810822148646082759110708590236243372097030127758714175370400577533310178835

15:

0.000002931796663246340125425992631775301191418078341574374387442988370470198789574807223832698756544038252

16:

0.000001253793750969930586964605206348193997730847263476479728160827654100091876779962330914878113124028959

17:

5.362125633443740052702565589398778192997447844064676329982266426514111512293486454541595755558282047e-7

18:

2.293287300515728443386931404793484964606665719068564893227677216571108402094779614342599095928196638e-7

19:

9.808124800989940117761878685673504826333187656318798972947288686244382352555202830057701906464031705e-8

20:

4.194854895721030216672516958988074617391798327273600024493305629088666429893230102430349408461265280e-8

Definition

The Airy zeta values are $Z_D(n)=\sum_{k=1}^{\infty}(-a_k)^{-n}$ for integers $n\geq2$, where $0>a_1>a_2>\cdots$ are the real zeros of the Airy function $\operatorname{Ai}$. [2]

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. [4]

(2)

$Z_D(2)=\kappa^2$, $Z_D(3)=\frac12-\kappa^3$, $Z_D(4)=\kappa^4-\frac{\kappa}{3}$, $Z_D(5)=\frac{5\kappa^2}{12}-\kappa^5$, and $Z_D(6)=\kappa^6-\frac{\kappa^3}{2}+\frac1{20}$. [2] [1]

(3)

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

(4)

If $q(z)=\operatorname{Ai}^{\prime}(z)/\operatorname{Ai}(z)=\sum_{m\geq0}q_m z^m$, then $q_0=-\kappa$ and $(m+1)q_{m+1}=\delta_{m,1}-\sum_{j=0}^{m}q_jq_{m-j}$ for $m\geq0$, where $\delta_{m,1}$ is $1$ for $m=1$ and $0$ otherwise. The zeta values are $Z_D(n)=(-1)^{n-1}q_{n-1}$ for $n\geq2$. This follows from the Airy equation $\operatorname{Ai}^{\prime\prime}(z)=z\operatorname{Ai}(z)$. [4]

Comments

(5)

The positive numbers $-a_k$, obtained from the Airy zeros, are the eigenvalues of $H=-d^2/dx^2+x$ on $x\geq0$, with the Dirichlet condition $u(0)=0$ 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 $Z_D(n)=\operatorname{Tr}(H^{-n})$, the trace of an integer power of the inverse operator. This is the odd-parity sector $Z^-(n)$ of [1].

(6)

The zero asymptotic $-a_k\sim[\frac{3\pi}{2}(k-\frac14)]^{2/3}$ implies that the sum converges for real order greater than $3/2$. [3] All entries are convergent sums; no analytic continuation is needed.

(7)

Some references use the negative-root convention $\sum_{k\geq1}a_k^{-n}=(-1)^n Z_D(n)$. The entries here use the positive spectrum, so every value is positive. The quantum-bouncer values in equations (4.12)–(4.16) of [2] use eigenvalues $(-a_k)/2$ and equal $2^n Z_D(n)$.

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)
kappa = B(3)**(QQ(1)/3) * B(QQ(2)/3).gamma() / B(QQ(1)/3).gamma()
q = [-kappa]
for m in range(19):
    convolution = sum((q[j]*q[m-j] for j in range(m+1)), B(0))
    q.append((B(1 if m == 1 else 0) - convolution) / B(m+1))
for n in range(2, 21):
    print(n, (-1)**(n-1) * q[n-1])

References

[1]

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

[2]

R. E. Crandall, On the quantum zeta function, Journal of Physics A 29 (1996), 6795–6816, Section 4, especially equations (4.12)–(4.22). (doi)

Links

[3]

NIST DLMF 9.9: Airy zeros and their asymptotics

[4]

NIST DLMF 9.2: Airy equation and initial values

Similar tables

Zeros of the Airy function $\operatorname{Ai}$ — The negative zeros whose absolute values form the Dirichlet spectrum.

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

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 Riccati arithmetic followed by evaluation in Arb balls. The Gamma arguments and the cube-root exponent were exact rationals. A 96-bit guard gives 429 working bits for 100 significant digits; the worst measured relative radius was less than 4.3e-122.

Every entry was recomputed at 629 bits and compared with an independent 700-bit Taylor-series division initialized by Arb evaluations of Ai(0) and Ai′(0). The low-order formulas for orders 2 through 6 were checked using the reflection-form Gamma expression for kappa. A separate sum over 64 numerically indexed, sign-bracketed Airy roots, with an explicit bound for the infinite tail, checked the scale and signs. That finite-root check does not certify the 100 digits or the root indices. The attached checks also test the recurrence, positivity, decrease, log convexity and containment in the stored decimal intervals.