Landau constants $G_n$

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Numbers

$n$

$G_n$

5/4

89/64

381/256

25609/16384

106405/65536

1755841/1048576

7207405/4194304

1886504905/1073741824

7693763645/4294967296

10:

125233642041/68719476736

11:

508710104205/274877906944

12:

33014475398641/17592186044416

13:

133748096600189/70368744177664

14:

2165115508033649/1125899906842624

15:

8754452051708621/4503599627370496

16:

9054883309760265929/4611686018427387904

17:

36559890613417481741/18446744073709551616

18:

590105629859261338481/295147905179352825856

19:

2379942639329101454549/1180591620717411303424

20:

153503933011295186253161/75557863725914323419136

21:

618542631098253663845669/302231454903657293676544

22:

9965857604589677025846929/4835703278458516698824704

23:

40128232690023504452613341/19342813113834066795298816

24:

10337821815102409932695615921/4951760157141521099596496896

25:

41600969157596518283705132645/19807040628566084398385987584

26:

669458243885941163049056453489/316912650057057350374175801344

27:

2692639901821666996862832256005/1267650600228229401496703205376

28:

173243054777620454995476049837345/81129638414606681695789005144064

29:

696503627371241284453523459678605/324518553658426726783156020576256

30:

11198692847520765866037293777995601/5192296858534827628530496329220096

31:

45006317827596641691308173501676685/20769187434139310514121985316880384

32:

185185505632327236351192343271564303049/85070591730234615865843651857942052864

33:

743999531711426604845251396017150821421/340282366920938463463374607431768211456

34:

11954590980457310705995730990034902456961/5444517870735015415413993718908291383296

35:

48015016436366078106067662654469152378885/21778071482940061661655974875633165533184

36:

3085199636196320192084492974801028517678241/1393796574908163946345982392040522594123776

37:

12388438731468203463060579867379460847173405/5575186299632655785383929568162090376495104

38:

198957335631722391580201050235374214600730105/89202980794122492566142873090593446023921664

39:

798722959317476289765401951307340740428871045/356811923176489970264571492362373784095686656

40:

205195440080875999820652281936008496258749301545/91343852333181432387730302044767688728495783936

41:

823601165637986556636978109264034769614102202205/365375409332725729550921208179070754913983135744

42:

13221661469281413789670669960676450656150928472505/5846006549323611672814739330865132078623730171904

43:

53058743966799515993261328320242274567978642845645/23384026197294446691258957323460528314494920687616

44:

3406524989418486814121830717672972978461416921171905/1496577676626844588240573268701473812127674924007424

45:

13668209853811537488078014372277432320662219938445645/5986310706507378352962293074805895248510699696029696

46:

219350548681618597366530869511504815256962001460064545/95780971304118053647396689196894323976171195136475136

47:

879983155857059923586964378685116116118361227508993405/383123885216472214589586756787577295904684780545900544

48:

903690882064688744579783649136653249259855433142579847345/392318858461667547739736838950479151006397215279002157056

49:

3624905852109051410229473025343512082774921575075627170005/1569275433846670190958947355801916604025588861116008628224

50:

58157541499435631090316731510517646027268559531441269212729/25108406941546723055343157692830665664409421777856138051584

51:

233253944266835926045352781034121271929902756381316335235965/100433627766186892221372630771322662657637687111424552206336

52:

14967410206194729435343210149560481800502946527771957290646049/6427752177035961102167848369364650410088811975131171341205504

53:

60023330553727818633881907740010650013263334906398980101308221/25711008708143844408671393477458601640355247900524685364822016

54:

962786997648034626699499151694916775337067929363175680382867561/411376139330301510538742295639337626245683966408394965837152256

55:

3860628081208619825406371577868011459317989785683138998321738645/1645504557321206042154969182557350504982735865633579863348609024

56:

990704547901359049315741083501139317963872656944418386964914010849/421249166674228746791672110734681729275580381602196445017243910144

57:

3972186680402148990047059278205883384545133162014654127898127921045/1684996666696914987166688442938726917102321526408785780068975640576

58:

63702309436180551123237675724852443118620492679373667200958382743745/26959946667150639794667015087019630673637144422540572481103610249216

59:

255388582296137109385203915093532062458447118264420415602213229797205/107839786668602559178668060348078522694548577690162289924414440996864

60:

16381331925586492637005040512123677546064750921865197711483224107116801/6901746346790563787434755862277025452451108972170386555162524223799296

61:

65668797142027846364690273524281051398682563403243667541600091998153005/27606985387162255149739023449108101809804435888681546220650096895197184

62:

1052959390227645436975941479944346516690069854139806216075329881724637969/441711766194596082395824375185729628956870974218904739530401550323154944

63:

4220729268343184005796564418910707173742315350074859845600640867893567501/1766847064778384329583297500742918515827483896875618958121606201292619776

64:

69295842681715168568523858431954362469107352267300180323697653888476816952009/28948022309329048855892746252171976963317496166410141009864396001978282409984

65:

277748236569395995525195691929603853335467307116590461536968736689563629783661/115792089237316195423570985008687907853269984665640564039457584007913129639936

66:

4452873220118369581286631099340931138914354203480925192025860426894357679719201/1852673427797059126777135760139006525652319754650249024631321344126610074238976

67:

17846569174493774000320556922103651026995958220920400909717699974119474880307429/7410693711188236507108543040556026102609279018600996098525285376506440296955904

68:

1144392418373552707923791569531537333340699516943455347083247182107413710132316081/474284397516047136454946754595585670566993857190463750305618264096412179005177856

69:

4586289871550185951512974693332149110180191952086065267820073304960246898426779949/1897137590064188545819787018382342682267975428761855001222473056385648716020711424

70:

73518175044100522399717698268983872855037248571170402648851753371729312946777784809/30354201441027016733116592294117482916287606860189680019559568902170379456331382784

71:

294615127288311340818136089889164031963516818060255853330841980907809093344921919861/121416805764108066932466369176469931665150427440758720078238275608681517825325531136

72:

75558411993356249893674395240437232004528560094866457584756160060421563303900908435041/31082702275611665134711390509176302506278509424834232340028998555822468563283335970816

73:

302773927771456186137836719582645215737157223015428706254125857586811361949851871380789/124330809102446660538845562036705210025114037699336929360115994223289874253133343883264

74:

4852910898511053478163658986265309486360475985909767835967710000140485424972918595983249/1989292945639146568621528992587283360401824603189390869761855907572637988050133502132224

75:

19445302497258137051067717717649190470490595007200712660413694948928542975170818100382021/7957171782556586274486115970349133441607298412757563479047423630290551952200534008528896

76:

1246625278528622326611704231704556238653323855918654429265073805478537209351762707119059369/509258994083621521567111422102344540262867098416484062659035112338595324940834176545849344

77:

4994894710158394401578028817628675858360628950018326039532578046781935771687013323431900501/2037035976334486086268445688409378161051468393665936250636140449354381299763336706183397376

78:

80050896654417821214549819186411925477708056606215368142972658917773681811868017870051633641/32592575621351777380295131014550050576823494298654980010178247189670100796213387298934358016

79:

320727220051998581356435004339655048710436071876323596042667243468267700424803221925733865189/130370302485407109521180524058200202307293977194619920040712988758680403184853549195737432064

80:

328954192407375682619865341615970759257117930275691920086279394180319322630842851828486535774761/133499189745056880149688856635597007162669032647290798121690100488888732861290034376435130433536

81:

1317908777543006970137015722598567196250782851953473047383974930139431061243944713690468812476269/533996758980227520598755426542388028650676130589163192486760401955554931445160137505740521734144

82:

21119605615140160388023043761542823757726734099769781070371678015465238166265347529171136218349329/8543948143683640329580086824678208458410818089426611079788166431288878903122562200091848347746304

83:

84609094455691579332937481715363849264797098331913995844908029292562787299799635028589896713997941/34175792574734561318320347298712833833643272357706444319152665725155515612490248800367393390985216

84:

5423245789545682219643587849504626817865297118967737759090569920635681575686530184569273474690223849/2187250724783011924372502227117621365353169430893212436425770606409952999199375923223513177023053824

85:

21725650397390724621999508213556390319008742217814015046949823091163212437933669047506634818833392421/8749002899132047697490008908470485461412677723572849745703082425639811996797503692894052708092215296

86:

348127022227197512574758497259597344181120956592734968027341223980770206513168612878105619537158986961/139984046386112763159840142535527767382602843577165595931249318810236991948760059086304843329475444736

87:

1394550868185006511979675363500804705073666912177872671243691148769386936483901155180490374996541575069/559936185544451052639360570142111069530411374308662383724997275240947967795040236345219373317901778944

88:

357522048001924717802083191344730545248393704747595874693663789402468217987451570105003712370743650108289/143343663499379469475676305956380433799785311823017570233599302461682679755530300504376159569382855409664

89:

1432133123154533086141521054620329242872400881227798533672399288179512971439038773970496867857771363073781/573374653997517877902705223825521735199141247292070280934397209846730719022121202017504638277531421638656

90:

22946486335349425714399786738411156117889014524409722283528077345480522815979928198763008824364768017461121/9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496

91:

91913952450410141073772733840545021086249742497557303280278203915734797055163383768958057020535874751825109/36695977855841144185773134324833391052745039826692497979801421430190766017415756929120296849762010984873984

92:

5890596605358796369782773021098108915171168391051356958786481122521684297433971300807150737185976598515387601/2348542582773833227889480596789337027375682548908319870707290971532209025114608443463698998384768703031934976

93:

23594453408488767713819912808493830438625684341992822309576024913404941098863852825219469624938642089262316029/9394170331095332911557922387157348109502730195633279482829163886128836100458433773854795993539074812127739904

94:

378018882634632740263505826258265105030604523889471122266184339629877098549814356735154165334397953159059697089/150306725297525326584926758194517569752043683130132471725266622178061377607334940381676735896625196994043838464

95:

1514084725365974702978335098147316147903897881903656042188784762509650601558555781358545431237990431989828193981/601226901190101306339707032778070279008174732520529886901066488712245510429339761526706943586500787976175353856

96:

6209835194710140733303795953633758409058383956019607596529526092235570849169907065724653136877080146390146849946801/2462625387274654950767440006258975862817483704404090416746768337765357610718575663213391640930307227550414249394176

97:

24871582441222768792030577459188128524465275128379250499337479302239931170746547727145251040552542185137161299787829/9850501549098619803069760025035903451269934817616361666987073351061430442874302652853566563721228910201656997576704

98:

398455935141231680006706375264516900675081173002022478961980565061846002692334023371091375540138686758903961282870889/157608024785577916849116160400574455220318957081861786671793173616982887085988842445657065019539662563226511961227264

99:

1595845626037167238808749062717339423923022319160336121742717893512878317227732867397660516058359515918229287360999181/630432099142311667396464641602297820881275828327447146687172694467931548343955369782628260078158650252906047844909056

100:

102262229964916618138612667958339840558657506398831274990957628181539156350669256543434906380322155662799274272273624809/40347654345107946713373737062547060536401653012956617387979052445947619094013143666088208645002153616185987062074179584

Definition

The Landau constant $G_n$ is the least $K$ such that $|\sum_{k=0}^{n}a_k|\leq K$ for every analytic function $f(z)=\sum_{k=0}^{\infty}a_kz^k$ on $|z|<1$ with $\sup_{|z|<1}|f(z)|\leq1$ [1].

Parameters

$n$

— largest retained Taylor degree ($n\geq0$)

Formulas

(1)

$\displaystyle G_n=\sum_{k=0}^{n}\frac{\binom{2k}{k}^{2}}{16^k}=\sum_{k=0}^{n}\left(\prod_{j=1}^{k}\frac{2j-1}{2j}\right)^2$ for $n\geq0$, where the empty product is $1$ [1] [2].

(2)

$4(n+1)^2(G_{n+1}-G_n)=(2n+1)^2(G_n-G_{n-1})$ for $n\geq1$, with $G_0=1$ and $G_1=5/4$.

Comments

(3)

The index $n$ is the largest retained Taylor degree, so the sum contains $n+1$ coefficients, including $a_0$. The coefficients may be complex; neither $f(0)=0$ nor univalence is imposed. The finite sum does not require a boundary value $f(1)$.

(4)

These coefficient-sum constants differ from the geometric Landau constant, which concerns disks contained in images of analytic maps. Landau–Kolmogorov constants bound intermediate derivatives on the real line.

(5)

The sequence $G_n$ is strictly increasing, and its positive successive increments decrease.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from math import comb

def landau(n):
    return sum((QQ(comb(2*k, k)**2) / QQ(16**k)
                for k in range(n + 1)), QQ(0))

values = [landau(n) for n in range(101)]

References

[1]

Y. Li, S. Liu, S. Xu and Y. Zhao, Asymptotics of Landau Constants with Optimal Error Bounds, Constructive Approximation 40 (2014), 281–305; section 1, equation (1.1). (arXiv) (doi)

Links

[2]

Chen and Choi: Inequalities and asymptotic expansions for the constants of Landau and Lebesgue, equation (1.1)

Similar tables

Lebesgue constants of the Fourier partial sums — sharp uniform-norm bounds for Fourier partial sums of continuous periodic functions, a different partial-sum problem

Data properties

Entries are of type: rational number

How well the digits are known: exact

Table is complete: no (it holds every Taylor degree $0\leq n\leq100$, covering the constant term and consecutive low and moderate truncation degrees; the cutoff is a practical reference choice for this discrete family, whose largest stored fraction has 240 characters)

How they were obtained:

Computed in the rational field $\mathbb Q$ by updating the squared odd/even product and its partial sum. Every value is an exact reduced fraction; no decimal rounding is used.

All 101 values were compared with an independent binomial-coefficient sum and with exact counts of returns of square-lattice walks through 200 steps. The walk check propagates endpoint counts rather than using binomial coefficients, and begins with known return counts $1,4,36$ at times $0,2,4$. The three-term recurrence, strict increase and decreasing increments were checked throughout the covered range. The attached checks can repeat these comparisons on the values read from the database.