Báez-Duarte coefficients

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Numbers

$k$

$c_k$

0.6079271018540266286632767792583658334261526480334792930736541913650387257734126471472556435537310257

-0.3160113010675635383604737147823166393383495187939643515614690011125909150262601752692587301038833894

-0.2569971117245732855793276438290749791516301696348011653528517886547180144812455728767835489832023829

-0.2109695312425177598281214806874545100187120829479615110858484168154811472124839999404786569488012856

-0.1748613476778813261894663800700917701897238848256097749825697173461550485615617640275380620249548561

-0.1463529100085022500614498216334450968599530914648930633987595860364607372959592563928080408944166319

-0.1236873465079117840280354947212747993688183572765952336784065766597483236540159375241279790580572496

-0.1055317454777628912155457163456384825945843929414686205116300046382315834620805412056787980093809527

-0.09087283359117716441180032497179109405914629015470178229576576562371624883506811736493346189765374683

-0.07893855041740799771548703426174051233850576854062951658071118499812973848748480089684112098321627261

10:

-0.06913906550510960397644752671867720388195476779188527086408521299100809779918372470036737638055661273

11:

-0.06102239553997541776358854305590496824984461547795555431894122041901712008477461163808202412817712490

12:

-0.05424098769819525922209410810620266451694534792392158120077013220393056319337539123813328465355081802

13:

-0.04852654199529754170513359093022288909664075670597746347809202636875449059438735901496945635342419556

14:

-0.04367102563751368786084243956690395286372496751142688711205583276913090265992487129184467938545354692

15:

-0.03951234271535708184154193773410007198928253542069257580220673768647229669232890464490244807474528730

16:

-0.03592350548575218054553354057950093097121278182164436801777712659929609155113054597795382701604228276

17:

-0.03280444080879194614988005354625472317671227960824185116787241957501162449489964288666620897090908870

18:

-0.03007578091924252747312483898358498933858949990876187035897500547166641590833927186678365245750145610

19:

-0.02767414953387737027948346066816771466670465379717656922895262318877239792581993793112881324613433555

20:

-0.02554857575961023247655369688834404356954264829950745157157547700734513310218127188144961052969354389

21:

-0.02365775945257207710959198908681927015775775736099869870482616164745307499569183616317898639388615033

22:

-0.02196798014472518399286085224682812811172626776588148502674204486125681806024243108150171977642160596

23:

-0.02045149307339450070945291948347745316297445018288554691898933367353253523129087157072352682169811439

24:

-0.01908529447447760874212432686420671730340449050702391278989100402788309060036914508950666875021313022

25:

-0.01785016732361390439341563248934803726923551908317392284123613302873118087379029785682829537336692120

26:

-0.01672994052566446017235628394160104460580268230583128964783598104796532070754861375680761080229462987

27:

-0.01571091095795920968759329199777395334260928483348103931004369546523234418758322281235758728322271847

28:

-0.01478139011486722779400486682755457762874319532489485519369209196453987080293436209288909971883424368

29:

-0.01393134639183992469255488612657007158797998584655905851509989388040450853400009687521036659252392104

30:

-0.01315212104526387655756809442285843483216097933364235939990380172309759592460453877846805394123229550

31:

-0.01243620113998297820791470970099314675112025814788219588687612200192094009338224921813114276159648562

32:

-0.01177703677688913910717376136194551936156163365285420007488198822299416024256182376332922926784143401

33:

-0.01116889289962889571816444174399265104094642354285685124923589161239512293332026958048294731574268810

34:

-0.01060672825334392274454955665579514373704700041283323108287663959382934681485761613987744101921474915

35:

-0.01008609579055392962418947000912381237364299803885393752420502542858166705784649100057208203720311011

36:

-0.009603060125856844462425996841264109738575568811946046189186790911537821840913011060012410209904459350

37:

-0.009154128634312588848378838892987718115314232007749644708848640184139456459588302093540142722847127395

38:

-0.008736193545063994368053115729249434076558038592503471835373612443456585901690003748683794915854200325

39:

-0.008346482959748257627334740671622070022892687259183493731795014058455540435484057659429338931571168803

40:

-0.007982519168072769990510239573466910914514951806483727453632563409336638928763815796555877966713700775

41:

-0.007642082973330460406395673099297708028394392758081617603414830965146692942727516645909165616160927385

42:

-0.007323183003297068059246673563763352743852730556829455294035587712059722398138836698300020433503659418

43:

-0.007024029185501996382075079459825747622482321192294966438279936157287561535442091833473317336631983343

44:

-0.006743009724348373730304516362224576196172774659751289332414261044521455931138850032988113333324013302

45:

-0.006478671041596955955647816725762654478398217652011156996285436929710080367148960440738845789113483157

46:

-0.006229700239359519186119461167132862716382576195009104246950786697315291744325328031384295886348102828

47:

-0.005994909722061906433524131114762607144964411334677189901162923178599269066683964065327416684638232315

48:

-0.005773223675460222878351102961492949913701870770896788781381388348501202651210233959225162268662873430

49:

-0.005563666150245198914006470715535119448549339022198332150178276047227713357764476863443090310513947583

50:

-0.005365350537733784116215667198393427183293774602373303035154712193841771005611778518465876400328454081

51:

-0.005177470257676662522000252935067319241840281067292745347267357255266109232974435551336260427144224046

52:

-0.004999290504880782506289206992979089574949043894474545691362603311466862765702112441961658051544625434

53:

-0.004830140923370030979204380113005137228930733674929709070993722897475444384581720126210442960877720482

54:

-0.004669409095122463561303810703835590103674541424743357519860866414561549552844275421120593362737264811

55:

-0.004516534745757100809355471304170880229396160563911851929227312935529365994557467716445884185582297568

56:

-0.004371004582465369259353309495583398450309286641071098475731006454836975407067931754954564749554858324

57:

-0.004232347690437566278613244628633972985246456016033750903583407551031252566179992444529467572608032882

58:

-0.004100131423375044360984685267862766531620706597986113245508843497316508411578224889796005638863069909

59:

-0.003973957731683506615590637294708687632005905357474074527724874490501576121776165050513314241848448974

60:

-0.003853459878835203391266646166904428380468950151521418723016816819555274956011620078813681020097109200

61:

-0.003738299502347679951670211593958671656673463277171276878824821802621721673418840861250508940982126402

62:

-0.003628163980999772101065619662071614964550963693440284891629462704379426991609463578489758121250624471

63:

-0.003522764074410769065914993963056272681270359915703195011511796833140577276703377587740601294155661806

64:

-0.003421831805043920107112908954456930290000973377703721047773699425848290353218657756636745698956550444

65:

-0.003325118556141808949043863661536109189363322171130272005179408301524019360556946406401715935569272236

66:

-0.003232393362126160512124145454756860759415699388785343805359153821621126098489713456835282664428519358

67:

-0.003143441370655133878433731866849382172441094834345000644234422866356209216066013331374252905423359792

68:

-0.003058062457875043476154843930312090342624515611437427912910915922153783753668979176112975380353143450

69:

-0.002976069980471484238984192129638320690420416832567630037948987571541195904327418269876864502201812712

70:

-0.002897289649951830133401489917452140398137162845971533431377884619738882866408651763574440639677380151

71:

-0.002821558516206921280593531137167410210748693451968858106218099337211700144529748073324468246236291264

72:

-0.002748724048830234221825434569171753174276569559534512425746063988352498490064810142237322717719566860

73:

-0.002678643305940267632161855600772475550598478119275420346366306408425566534222252929651315924523082517

74:

-0.002611182181375673031849703158796621314196449218818354778753617074489513387369289531443040882463802426

75:

-0.002546214722129730108269605707092486923228911155064730374795087518067564991838926551098090197754108556

76:

-0.002483622508775894615713130831762302119631406639095481982089501240737024463318024481220976764914834859

77:

-0.002423294092422290076345725404054873963023437374337534895563768860708082453673510404601781255153319197

78:

-0.002365124482431550531145543989491610360083394783754196688582292107177795436554364007010552442004778214

79:

-0.002309014679763360090906870752148527588359603404863182721196122623158963172048409796188835215822163881

80:

-0.002254871251349195937113760886583953624074238976155703765712826511354549398958600015907656037613740511

81:

-0.002202605941399946811708764054580328526950825244013993065640678467756178947140983036274091244543239591

82:

-0.002152135315984123289006641474554137479908892301685100730601415716716196295868482757974208391148873930

83:

-0.002103380437603377846537229159840115516701686281324262424596601862049939277602400012765800644257034046

84:

-0.002056266566838391498586334756542180822865629397489668573919952374570972901133761922860185206333266871

85:

-0.002010722888446623916603847570088369195621904412507708089145373225957823155082336792633953675354871258

86:

-0.001966682259568187759887460538404553082867312709452360337383737788302236207904196872819436312439641994

87:

-0.001924080977940939153705247219117111832290208449478294584013741865740291875982916013298235952926168767

88:

-0.001882858568244095961743064297491370992585358764382109393141550372512309421352759468020181533265987739

89:

-0.001842957584884250993713506316911398451176731523556925800149247164088086923183581703961109146794808980

90:

-0.001804323429711155035658393136245038395895212702168016835717466979225909071508131136662540458561595315

91:

-0.001766904183305428067128637760156273497698608995891626071512295516997815314237105413586473823914513608

92:

-0.001730650448618480069988278836258555480772561991943671232395684213459160813526592435082976437797519551

93:

-0.001695515205868219063931057974016202454396189781552653953470096005857796730581939601542290719062457043

94:

-0.001661453677704222851013655887574052871667254967182947797999504330596396095169705622961230811135330622

95:

-0.001628423203754400647209243117889176384355118548560605018257480183993852164451023144534062458341149007

96:

-0.001596383123753058719921518141738032183252201838165496743321510959320515248146076332337678563575786693

97:

-0.001565294668528854932082841073785320180543404252355547998326774395287835380258355918656344619411365816

98:

-0.001535120858201398491807785759008475174576015155631240812242764696209120817109339335286989774559741629

99:

-0.001505826406998128347953381576125667450977023552176784761966527780474866253090698746184545998869250321

100:

-0.001477377634159391602150758922435834445647783886480725749705048037236708217565105867734480229790783576

101:

-0.001449742380450058144903514714028964023381602264562720613696699007614809948503046404600800783954947972

102:

-0.001422889929841186556423915320029970649998418575496801242222415067603875877963166779423186563534556267

103:

-0.001396790935965765991024473101790689685249374005399985971926386796567232280000812246286470661956667483

104:

-0.001371417352988903700255540468228220108614282098872126428329726306442604487462732516451855025471755238

105:

-0.001346742370565456953501939296220096084203496063121004702021859982857109178159757308086742538852240724

106:

-0.001322740352587420947631066684116406621712723788744139192631106894946098758680270375575671816850712866

107:

-0.001299386779449736545630200957314019808445744917249179957182018747110052316742507808319915334213672192

108:

-0.001276658193586890018919356118718722758715839559715864994960479470899560652282737246376275820967926232

109:

-0.001254532148054023364587431623431358961079942839556684046993476158576194523425611231072569600789470460

110:

-0.001232987157945509346044008577323745391731347327015744542804139859862412421399206876065177882029079461

111:

-0.001212002654461293837937019663913618130209931204713590417813210879924387529015368906720499851811471154

112:

-0.001191558941446968606548761578741429866483304008459228620872178355297490334883012490483286815087370158

113:

-0.001171637154247687872627137108213103555516187089007042747455159067434558940698863756260538870869769600

114:

-0.001152219220728840113315287902950212641739980887704427011721154392128579340144164746542086822449045598

115:

-0.001133287824327973604048091895614878694924714254069448852152041117906607795158501702247732283952309343

116:

-0.001114826369012975926894628838535264641245291481839560843348963537775522838659366916914408901466289648

117:

-0.001096818946031036216487263301624035214386522958159803676567828833225630401238256530902189317152231989

118:

-0.001079250302341574304468028517715656154496342452845696237989596301429485246974995268095894722033244977

119:

-0.001062105810634192404581888891813048323621643148841921679850137551801739946438762588430863357691033643

120:

-0.001045371440839872203927140210597139572552002238958666764516062237575287106988953337025835398129607794

121:

-0.001029033733050174299895216402395829753206547466708429361050918213798815532698663390411210542092641447

122:

-0.001013079771765161349800231884852869452292322253310473593004547407621733564223307557872006993222315477

123:

-0.0009974971613962178113248237918939937803378428932017268135021207226507383914593846679351954489146716043

124:

-0.0009822740029549284452779087881480481707745557090898861948530639932674891664409468720994569401142626037

125:

-0.0009673988718637501538170347454051255315747245182137273547667303270019500596095326575010847230206093951

126:

-0.0009528607968284077770333607160115582948754445172092487474936130678991938746880154241541598387161447844

127:

-0.0009386492397158004437609786871734938213769801391963474541676169835571025235067427340292124551929423878

128:

-0.0009247540763847534431453712341740767460436309000612274280404450832863059600119277865885799038592144281

129:

-0.0009111655784202204386072171103376941335111479344885251397969849581953826775648806718149707635330676674

130:

-0.0008978743957245582516870423426320706780661801270295539809247005075872887682162451750315850358771319191

131:

-0.0008848715399222847726926677781208189207196377330180936303078622551320945625528462586703227077204859639

132:

-0.0008721483685373107784005186740175353177209051404005676857022846869645588254775891083994155134429833972

133:

-0.0008596965699040273810122999564810240757431680232489970923759328095827021044698915507545911188425216694

134:

-0.0008475081487758494108007812664973831876213571279902667681956576123986100579821484608090549024284401801

135:

-0.0008355754125968764531931839932805210897868495702954843406604456674156388416316811300183508521440510610

136:

-0.0008238909584042512000857555034879915591542160618292961826280457114066212865142719525398209812678953273

137:

-0.0008124476603305815531471309536663035213588154740377389951393170233791796874505391345399140104135320970

138:

-0.0008012386576774596343728208930584071564871883524684593024268332751240433115316841816695971693069092539

139:

-0.0007902573435326675283118137093100816914820087588325004266011683211869750488796760652707000453200600683

140:

-0.0007794973539051152397759719580220160096763102955361318350069089950972022953437263597334184059368662401

141:

-0.0007689525573529191679464254810903239774051364351021688107220616471812859158042719084361144077968177583

142:

-0.0007586170450813067602903568026676867220675281600037517565586358892984459989771220725186150511100400842

143:

-0.0007484851214882316068515540869333501817952023357705033176065711306945573737433452815235639759430327132

144:

-0.0007385512951367091306554183442386260195923602020037921166470338268873213501944371848617168857721183205

145:

-0.0007288102701339417253657068485427622222105639571602060276005997801753383556682975543225335877782913136

146:

-0.0007192569378982986856576519328105460487586995268541082988077624087302856665756401623629939309507398266

147:

-0.0007098863692961551157440702849689892328953806904342994457002924930971746013792476965503210015986151551

148:

-0.0007006938071314793277690185821161444822914623398978533146423209007024149850217966188593435014060327864

149:

-0.0006916746589718938299915459930612744232099344380821022760691933037174915417090554881827329439610821259

150:

-0.0006828244902957243020330148317145093590864503310684860740334887175815603108815734787897660093118993457

151:

-0.0006741390179452971124746454018214417512823456964528219180496112843442075947177150435842496999766057657

152:

-0.0006656141038724518378302248194849716162406215988759505547247607101444286796954828086583027595938669524

153:

-0.0006572457491629035361254606086321752069234246817698633068457424748698978647644568954522160540797636980

154:

-0.0006490300883267226407887903717334151223732747396493637432833464154658258215602647605124484685915658010

155:

-0.0006409633838428005030668303570021412947008344790446011596118799466280495554001770470262181489347041821

156:

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Definition

The Báez-Duarte coefficients are $c_k=\sum_{j=0}^{k}(-1)^j\binom{k}{j}/\zeta(2j+2)$ for integers $k\geq0$ [1].

Parameters

$k$

— Coefficient index ($k\geq0$)

Formulas

(1)

$c_0=6/\pi^2$, $c_1=6/\pi^2-90/\pi^4$, and $c_2=6/\pi^2-180/\pi^4+945/\pi^6$ [3], equation 25.6.1.

(2)

$\displaystyle \frac{1}{\zeta(2k+2)}=\sum_{j=0}^{k}(-1)^j\binom{k}{j}c_j$ for integers $k\geq0$, by binomial inversion.

(3)

$\displaystyle c_k=\sum_{j=0}^{k}(-1)^j\binom{k}{j}\frac{2(2j+2)!}{(2\pi)^{2j+2}|B_{2j+2}|}$, where $B_n$ denotes a Bernoulli number [3], equation 25.6.2.

(4)

$\displaystyle \sum_{k=0}^{\infty}c_k\frac{x^k}{k!}=e^x\frac{R(x)}{x}$, 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 $x$ [2], equations (2) and (10).

Comments

(5)

The Riemann hypothesis is equivalent to $c_k=O(k^{-3/4+\varepsilon})$ for every $\varepsilon>0$ as $k\to\infty$ [1], Theorem 1.1. These finite coefficient computations are unconditional and do not establish the asymptotic bound.

(6)

The leading constants of zeta moments also use the symbol $c_k$ for a different sequence. The Keiper-Li coefficients arise from a different transform and give Li's positivity criterion.

(7)

Here $\zeta$ is the ordinary Riemann zeta function. The entries retain the signs in the definition. The constant $\pi$ is positive. The convention for Bernoulli numbers used in (3) is $t/(e^t-1)=\sum_{n\geq0}B_nt^n/n!$, so $B_1=-1/2$.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.real_arb import RealBallField
from math import comb

def c(k, digits=100):
    F = RealBallField(numberdb.bits(digits, losing=k+80))
    return sum(((-1)**j * comb(k,j) / F(2*j+2).zeta()
                for j in range(k+1)), F(0))

[c(k) for k in range(257)]

References

[1]

L. Báez-Duarte, A new necessary and sufficient condition for the Riemann hypothesis (2003), definition (1.1) and Proposition 2.1. (arXiv)

[2]

J. Cisło and M. Wolf, On the Riesz and Báez-Duarte criteria for the Riemann Hypothesis (2008), equations (2), (4), and (10). (arXiv)

Links

[3]

DLMF 25.6: integer arguments of the Riemann zeta function

Similar tables

Values of the Riemann zeta function at rational numbers — The reciprocals of the even positive zeta values supply the finite-difference inputs defining $c_k$.

Bernoulli numbers — Euler's even-zeta identity gives the rational coefficients in (3).

Keiper-Li coefficients — Li's criterion is a different equivalent formulation of the Riemann hypothesis, using nonnegativity of another coefficient sequence.

Values of the Riesz function — The Riesz function satisfies $R(x)=xe^{-x}\sum_{k=0}^{\infty}c_k x^k/k!$. Its integer arguments support comparison of $R(k)/k$ with $c_k$; the quotient at $x=0$ has the continuous limit $c_0$.

Alternating sum of the Báez-Duarte coefficients — The constant is the ordinary convergent alternating sum $S=\lim_{K\to\infty}\sum_{k=0}^{K}(-1)^k c_k$, starting at $k=0$ and retaining the signs of these coefficients.

Data properties

Entries are of type: real number

digits: 100 (Unknown key)

How well the digits are known: proven

Table is complete: no (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)

How they were obtained:

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.

An independent calculation builds exact 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 (3) 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.