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{"Title": "Airy zeta values", "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 $\\operatorname{Ai}$. CITE{Crandall}", "Parameters": {"n": {"type": "Z", "display": "$n$", "title": "inverse-power order", "constraints": "$n\\geq2$"}}, "References": {"Voros": {"bib": "A. Voros, Airy function (exact WKB results for potentials of odd degree), Journal of Physics A 32 (1999), 1301\u20131311, Section 3.", "arxiv": "math-ph/9811001", "doi": "10.1088/0305-4470/32/7/020"}, "Crandall": {"bib": "R. E. Crandall, On the quantum zeta function, Journal of Physics A 29 (1996), 6795\u20136816, Section 4, especially equations (4.12)\u2013(4.22).", "doi": "10.1088/0305-4470/29/21/014"}}, "Links": {"DLMFzeros": {"title": "NIST DLMF 9.9: Airy zeros and their asymptotics", "url": "https://dlmf.nist.gov/9.9"}, "DLMFinitial": {"title": "NIST DLMF 9.2: Airy equation and initial values", "url": "https://dlmf.nist.gov/9.2"}}, "Tags": ["special values", "special functions", "analysis", "physics"], "Data properties": {"type": "R", "complete": "no", "complete-note": "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", "rigour": "proven", "digits": "100", "rigour details": "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\u2032(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."}, "Display properties": {"number-header": "$Z_D(n)$"}, "Numbers": {"2": "0.5314572319609994528671249327835975128793762613598310302129500019359222956528578262843381519744913357", "3": "0.1125617612151145794304351152459810519503961166363151095605355424412345609571039396593371385193231989", "4": "0.03944307842123858454386231559273186269551737139039066301468634771716695964798177446256180160638766378", "5": "0.01553365937662315960105326698601338207533195387154164863322026920862587812029184977321978498881146467", "6": "0.006389269480291183085985031797193603176325864055526664641413635547275766261945530323301383069312594316", "7": "0.002679136468944483094339090139571816658235339260887016458460696566390563589062212139425800604794000764", "8": "0.001133964981406675717919109806835643688471429708423116819424855547821787518793036055882728723770737149", "9": "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"}, "Comments": {"comment-spectrum": "The positive numbers $-a_k$, obtained from the HREF{Zeros_of_the_Airy_function_Ai}[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 CITE{Voros}.", "comment-convergence": "The zero asymptotic $-a_k\\sim[\\frac{3\\pi}{2}(k-\\frac14)]^{2/3}$ implies that the sum converges for real order greater than $3/2$. CITE{DLMFzeros} All entries are convergent sums; no analytic continuation is needed.", "comment-sign": "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)\u2013(4.16) of CITE{Crandall} use eigenvalues $(-a_k)/2$ and equal $2^n Z_D(n)$."}, "Formulas": {"formula-kappa": "Write $\\kappa=-\\operatorname{Ai}^{\\prime}(0)/\\operatorname{Ai}(0)=3^{1/3}\\Gamma(2/3)/\\Gamma(1/3)$, where $\\Gamma$ is the HREF{Values_of_the_Gamma_function_at_rational_numbers}[Gamma function]. CITE{DLMFinitial}", "formula-low": "$Z_D(2)=\\kappa^2$, $Z_D(3)=\\frac12-\\kappa^3$, $Z_D(4)=\\kappa^4-\\frac{\\kappa}{3}$, $Z_D(5)=\\frac{5\\kappa^2}{12}-\\kappa^5$, and $Z_D(6)=\\kappa^6-\\frac{\\kappa^3}{2}+\\frac1{20}$. CITE{Crandall} CITE{Voros}", "formula-log": "For $|z|<-a_1$, the logarithm analytic near $z=0$ and equal to zero there satisfies $\\log\\!\\left(\\operatorname{Ai}(z)/\\operatorname{Ai}(0)\\right)=-\\kappa z+\\sum_{n=2}^{\\infty}\\frac{(-1)^{n-1}}{n}Z_D(n)z^n$. CITE{Voros}", "formula-riccati": "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 $\\operatorname{Ai}^{\\prime\\prime}(z)=z\\operatorname{Ai}(z)$. CITE{DLMFinitial}"}, "Similar tables": [{"table": "HREF{Zeros_of_the_Airy_function_Ai}", "relation": "The negative zeros whose absolute values form the Dirichlet spectrum."}, {"table": "HREF{Values_of_the_Airy_function_of_the_first_kind}", "relation": "Values of the function whose logarithmic expansion yields these spectral moments."}], "Keywords": ["Airy zeta", "Airy spectral zeta", "quantum bouncer", "Dirichlet Airy spectrum", "inverse spectral moments"], "Programs": {"program-sage": {"language": "Sage", "code": "import numberdb.sage as numberdb\nfrom sage.rings.rational_field import QQ\nfrom sage.rings.real_arb import RealBallField\n\nB = RealBallField(429)\nkappa = B(3)**(QQ(1)/3) * B(QQ(2)/3).gamma() / B(QQ(1)/3).gamma()\nq = [-kappa]\nfor m in range(19):\n convolution = sum((q[j]*q[m-j] for j in range(m+1)), B(0))\n q.append((B(1 if m == 1 else 0) - convolution) / B(m+1))\nfor n in range(2, 21):\n print(n, (-1)**(n-1) * q[n-1])\n"}}}