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{
"Title": "Báez-Duarte coefficients",
"Definition": "The Báez-Duarte coefficients are $c_k=\\sum_{j=0}^{k}(-1)^j\\binom{k}{j}/\\zeta(2j+2)$ for integers $k\\geq0$ CITE{BD}.",
"Parameters": {
"k": {
"type": "Z",
"title": "Coefficient index",
"display": "$k$",
"constraints": "$k\\geq0$"
}
},
"Keywords": [
"Baez-Duarte",
"sequential Riesz criterion"
],
"Tags": [
"number theory",
"L-function",
"sequence"
],
"Links": {
"DLMF": {
"title": "DLMF 25.6: integer arguments of the Riemann zeta function",
"url": "https://dlmf.nist.gov/25.6"
}
},
"References": {
"BD": {
"bib": "L. Báez-Duarte, A new necessary and sufficient condition for the Riemann hypothesis (2003), definition (1.1) and Proposition 2.1.",
"arxiv": "math/0307215"
},
"CW": {
"bib": "J. Cisło and M. Wolf, On the Riesz and Báez-Duarte criteria for the Riemann Hypothesis (2008), equations (2), (4), and (10).",
"arxiv": "0807.2971"
}
},
"Display properties": {
"number-header": "$c_k$"
},
"Data properties": {
"type": "R",
"digits": "100",
"rigour": "proven",
"complete": "no",
"complete-note": "it holds the consecutive coefficient orders $0\\leq k\\leq256$, preserving the initial segment needed for binomial inversion and reciprocal-zeta expansions; the noncanonical cutoff balances finite-difference cancellation against 100-digit reference coverage",
"rigour details": "Successive differences of reciprocal even zeta values are evaluated with Arb real balls at 669 working bits. The error bounds propagate through every subtraction; all 257 computed balls retain at least 398 relative accuracy bits. Each entry is written to 100 significant digits, with the last-place interval enclosing the result.\n\nAn independent calculation builds exact HREF{Bernoulli_numbers}[Bernoulli numbers] through $B_{514}$ from their generating-function recurrence, encloses $\\pi$ using Machin's arctangent series with explicit alternating-series remainder bounds, and evaluates the rational polynomial in CITE{formula-bernoulli} by Horner's rule at 883 bits. Every independent enclosure is contained in its written 100-digit interval. The first three closed forms, binomial inversion, and the exponential-generating-function coefficients through order 256 were checked. Binomial inversion on the stored decimal intervals retains at least 60 relative bits because that transform amplifies their uncertainty. The attached checks reproduce these tests, including finite-width and deliberate-error controls."
},
"Numbers": {
"0": "0.6079271018540266286632767792583658334261526480334792930736541913650387257734126471472556435537310257",
"1": "-0.3160113010675635383604737147823166393383495187939643515614690011125909150262601752692587301038833894",
"2": "-0.2569971117245732855793276438290749791516301696348011653528517886547180144812455728767835489832023829",
"3": "-0.2109695312425177598281214806874545100187120829479615110858484168154811472124839999404786569488012856",
"4": "-0.1748613476778813261894663800700917701897238848256097749825697173461550485615617640275380620249548561",
"5": "-0.1463529100085022500614498216334450968599530914648930633987595860364607372959592563928080408944166319",
"6": "-0.1236873465079117840280354947212747993688183572765952336784065766597483236540159375241279790580572496",
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"8": "-0.09087283359117716441180032497179109405914629015470178229576576562371624883506811736493346189765374683",
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"254": "-0.0002429844098019314992153345028501694589059032725085789079023087635461465076998865320889522334648316641",
"255": "-0.0002411323784405373952910221270935809923780904426846696794988295277089530632382557585977597791916477637",
"256": "-0.0002393025088236655493611900775631511967911467448350320151249535273613963538607937510228905485002552559"
},
"Comments": {
"criterion": "The Riemann hypothesis is equivalent to $c_k=O(k^{-3/4+\\varepsilon})$ for every $\\varepsilon>0$ as $k\\to\\infty$ CITE{BD}, Theorem 1.1. These finite coefficient computations are unconditional and do not establish the asymptotic bound.",
"notation": "The HREF{Leading_constants_of_the_moments_of_the_Riemann_zeta_function}[leading constants of zeta moments] also use the symbol $c_k$ for a different sequence. The HREF{Keiper-Li_coefficients}[Keiper-Li coefficients] arise from a different transform and give Li's positivity criterion.",
"conventions": "Here $\\zeta$ is the ordinary HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers}[Riemann zeta function]. The entries retain the signs in the definition. The constant $\\pi$ is positive. The convention for HREF{Bernoulli_numbers}[Bernoulli numbers] used in CITE{formula-bernoulli} is $t/(e^t-1)=\\sum_{n\\geq0}B_nt^n/n!$, so $B_1=-1/2$."
},
"Formulas": {
"formula-first": "$c_0=6/\\pi^2$, $c_1=6/\\pi^2-90/\\pi^4$, and $c_2=6/\\pi^2-180/\\pi^4+945/\\pi^6$ CITE{DLMF}, equation 25.6.1.",
"formula-inverse": "$\\displaystyle \\frac{1}{\\zeta(2k+2)}=\\sum_{j=0}^{k}(-1)^j\\binom{k}{j}c_j$ for integers $k\\geq0$, by binomial inversion.",
"formula-bernoulli": "$\\displaystyle c_k=\\sum_{j=0}^{k}(-1)^j\\binom{k}{j}\\frac{2(2j+2)!}{(2\\pi)^{2j+2}|B_{2j+2}|}$, where $B_n$ denotes a HREF{Bernoulli_numbers}[Bernoulli number] CITE{DLMF}, equation 25.6.2.",
"formula-egf": "$\\displaystyle \\sum_{k=0}^{\\infty}c_k\\frac{x^k}{k!}=e^x\\frac{R(x)}{x}$, where the Riesz function is $R(x)=x\\sum_{j=0}^{\\infty}(-1)^jx^j/(j!\\zeta(2j+2))$. The quotient at $x=0$ is interpreted by continuity as $\\lim_{x\\to0}R(x)/x=1/\\zeta(2)$; the identity holds for every complex $x$ CITE{CW}, equations (2) and (10)."
},
"Similar tables": [
{
"table": "HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers}[Values of the Riemann zeta function at rational numbers]",
"relation": "The reciprocals of the even positive zeta values supply the finite-difference inputs defining $c_k$."
},
{
"table": "HREF{Bernoulli_numbers}[Bernoulli numbers]",
"relation": "Euler's even-zeta identity gives the rational coefficients in CITE{formula-bernoulli}."
},
{
"table": "HREF{Keiper-Li_coefficients}[Keiper-Li coefficients]",
"relation": "Li's criterion is a different equivalent formulation of the Riemann hypothesis, using nonnegativity of another coefficient sequence."
}
],
"Programs": {
"program-sage": {
"language": "Sage",
"code": "import numberdb.sage as numberdb\nfrom sage.rings.real_arb import RealBallField\nfrom math import comb\n\ndef c(k, digits=100):\n F = RealBallField(numberdb.bits(digits, losing=k+80))\n return sum(((-1)**j * comb(k,j) / F(2*j+2).zeta()\n for j in range(k+1)), F(0))\n\n[c(k) for k in range(257)]"
}
}
}