Values of the Brjuno function

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Numbers

$x$

$B(x)$

$\frac{- 1+\sqrt{5}}{2}$:

1.259828913794410219858429911324809416483489854753852847937784883250947320738747613319876974555610157

comment: The argument is the reciprocal of the golden ratio, with continued fraction $[0;\overline{1}]$.

$\frac{- 2+\sqrt{8}}{2}$:

1.504598827159773538626629405230360311678855640608202663652010384749547188769040268301375965145882671

comment: The argument is the reciprocal of $1+\sqrt2$, with continued fraction $[0;\overline{2}]$.

$\frac{- 2+\sqrt{12}}{2}$:

1.431123181596072700185137618996141681247017657845543905090233091628383413090084567396396427001356104

$\frac{- 2+\sqrt{12}}{4}$:

1.528879979151354034247119727823971752108026436499680148681383237285622908155260211140530718758106818

$\frac{- 3+\sqrt{13}}{2}$:

1.713599354733757142283113582199163219852328125892443282296272854247834904950969666976002758186175768

$\frac{- 3+\sqrt{17}}{2}$:

1.451633307614547259695014480851439078804653701182863840688740692929697997579027542558238430646289435

$\frac{- 1+\sqrt{17}}{4}$:

1.557438294473179042469845334970110417306243746857397155756789633154633321505284216116005177682578764

$\frac{- 3+\sqrt{17}}{4}$:

1.677781016665859071886656607578976103617059946135728158606065286376715690530068852438812161557628580

$\frac{- 4+\sqrt{20}}{2}$:

1.889743370691615329787644866987214124725234782130779271906677324876420981108121419979815461833415235

$\frac{- 3+\sqrt{21}}{2}$:

1.628544344062572220905200485243330543543536467719521230662402375686942301343320671229429220111594832

$\frac{- 3+\sqrt{21}}{6}$:

1.762254878998798359776628548674200710770495230669108255994717130029959580784741971113376114440881842

$\frac{- 4+\sqrt{24}}{2}$:

1.635894703730683233998041334860078208699004807082665487675643257365333211371946191308521412971831813

$\frac{- 4+\sqrt{24}}{4}$:

1.860448369860695572235165316651317642459602748032769907813835690307256773270453629963762602953739821

$\frac{- 4+\sqrt{28}}{2}$:

1.431687751400907682968657331517715752415602235663127600734987142116566059093566031961091769636867015

$\frac{- 2+\sqrt{28}}{4}$:

1.712435349462716452676414596630354378484880298502164828285366615637893276224716503276344412197317504

$\frac{- 2+\sqrt{28}}{6}$:

1.539829525163240325596657480387500854774503057910321854086307644114364473107000665879141118934073375

$\frac{- 4+\sqrt{28}}{6}$:

1.844124519546306206847643832549522304811082178017828393120294664890529522750789064813193654550113911

$\frac{- 5+\sqrt{29}}{2}$:

2.040122922325111958684829636677033187429072993071313912322040049402857634449647789853827033244273030

$\frac{- 4+\sqrt{32}}{2}$:

1.801730139215244110094599266631728059584955522369620635567875070290287042607797722949654307567576460

$\frac{- 4+\sqrt{32}}{8}$:

1.947671297279147705529735966167573067086420941682384985014979608492196847463060372315921840949066908

$\frac{- 5+\sqrt{33}}{2}$:

1.558460681465136992380680698124171659729290035318912416998821763971730476195299211493186936415198457

$\frac{- 3+\sqrt{33}}{4}$:

1.532054338061401953034064789072763613051370036029288536251198137789206243293561448641235813125453586

$\frac{- 5+\sqrt{33}}{4}$:

1.971345548110835273244361394558596077076888170781016874901200247158873946482201650838503535974810193

$\frac{- 3+\sqrt{33}}{6}$:

1.683884626729728057214146274856366389390857744374819378796756638334072771715667963415831198183042101

$\frac{- 5+\sqrt{37}}{2}$:

1.623043382555102346440049005020412715767492893607199588068728225437822488581531619687558081542131919

$\frac{- 1+\sqrt{37}}{6}$:

1.864512018943497600723823709813141514189803830687946089435388802855831359597103233012358075029144895

$\frac{- 5+\sqrt{37}}{6}$:

2.005138888801717574988037715295665593253766952233835874806047901056564408873329163254835418295762432

$\frac{- 6+\sqrt{40}}{2}$:

2.170703075195020294257618179230824439014497038219790885218226153289307822111606412981547384464137277

$\frac{- 4+\sqrt{40}}{4}$:

1.349178084261476876428528627296958977697278794499313154738040047656516514466332950926864684645209062

$\frac{- 2+\sqrt{40}}{6}$:

1.387641716698451873679154650932897391076541275813754745777507882634979854730340098298827756385052007

$\frac{- 4+\sqrt{40}}{6}$:

1.470937224193448339954573007638939517132367714166717545639912523059943472898722591124116049535422820

$\frac{- 5+\sqrt{41}}{2}$:

1.461533404988976752895844659892400531885698107780465690816673252213800655099745538392226316781553365

$\frac{- 3+\sqrt{41}}{4}$:

1.860677710110214971899548332316764005136487304503314150659829264224471452244899735312185473246233967

$\frac{- 5+\sqrt{41}}{4}$:

1.700283510925932353686231585229122507545176948838474203312046159642750255723378095314914012429766100

$\frac{- 3+\sqrt{41}}{8}$:

1.578032141936607081042148687238084584664776026639773730259044529728628245390254530020625536039713471

$\frac{- 5+\sqrt{41}}{8}$:

1.997079311253928147118784661829852249115703048170556644126733637560768687892149163618184154197685748

$\frac{- 6+\sqrt{44}}{2}$:

1.825121251848566282448628886095753637880515661379894077552091566542806951270718548843437969868418171

$\frac{- 6+\sqrt{44}}{4}$:

2.132124330213031717917357624979940376853312593435264457251606199252787012390031013366989327457362161

$\frac{- 5+\sqrt{45}}{2}$:

1.951786478068141126063619118046757727914637528745051367568005801722017958613887414032450392804228569

$\frac{- 5+\sqrt{45}}{10}$:

2.100547540059766309435123533162754665964126131880130971976551609742058300328747383316915440664686431

$\frac{- 6+\sqrt{48}}{2}$:

1.760161842926981067562946715916049740628841351450092500410962737821246675857402291877075990359697579

$\frac{- 6+\sqrt{48}}{6}$:

2.138562025994291090440193245511796931927299155860579395200081419839212927349594002115808387456402102

$\frac{- 6+\sqrt{52}}{2}$:

1.423175385183992313590807739230631227897681219276271498869201204966557487135661950281387191231885923

$\frac{- 2+\sqrt{52}}{6}$:

1.967773700642119888450737967805128045568744374789540477248658086518348265090502133256326412016595646

$\frac{- 4+\sqrt{52}}{6}$:

1.678265642213899965629075488775966025724297454210924672644692678734196282572975506963028003254989472

$\frac{- 2+\sqrt{52}}{8}$:

1.521851882403691313749548374066745905070603593135158899888636468607750272388657534545580635501257014

$\frac{- 6+\sqrt{52}}{8}$:

2.103361815273834812539490324108731160264426895361763388452401939611721149619531330975965033913575070

$\frac{- 7+\sqrt{53}}{2}$:

2.285867520718904027320764644576624862842952175660261695441256833902634965186833785696843631612259310

$\frac{- 6+\sqrt{56}}{2}$:

1.599441437679264793986396242328581823773778546347632730762468416588279905065677057035257173276088477

$\frac{- 4+\sqrt{56}}{4}$:

2.006719384376595511463555230870457598499455979028186939218547906477145001166501937506620436668866503

$\frac{- 4+\sqrt{56}}{10}$:

1.753604259229992786201864718669685115062265464577341376137603444063185870651422580952733843745137083

$\frac{- 6+\sqrt{56}}{10}$:

2.145553309170627307009266295926103473125730671507476574575215308251592028254327492502781070679479718

$\frac{- 7+\sqrt{57}}{2}$:

1.681337630110131855383193049140250072677199501962439929833370525768939063068748818035004931203510201

$\frac{- 5+\sqrt{57}}{4}$:

1.418799375504319565415982776882059292553336735243213286474744543535293375281543332162489379825754152

$\frac{- 7+\sqrt{57}}{4}$:

2.215546765561388467663028707027773031723907587073075248979629929811874708323521893157671422373410804

$\frac{- 3+\sqrt{57}}{6}$:

1.679210287614831498901171123712722283022107398628266327675285257459963146203682189881341359231481925

$\frac{- 3+\sqrt{57}}{8}$:

1.519366796062970838888308986594489436202159349978346156524474014445045553108203712774947454322768198

$\frac{- 5+\sqrt{57}}{8}$:

1.849572791434821361133339858339126790027016104910129619768248625207528802761510840706242436378159885

$\frac{- 6+\sqrt{60}}{2}$:

2.083201237232838713430804130475006111126971953866647574017046781071668353463035373601449668154141431

$\frac{- 6+\sqrt{60}}{4}$:

1.566826683831817087746912946075559618293666523286127894386562782945427684910809555550135546726367864

$\frac{- 6+\sqrt{60}}{6}$:

1.690388955628052495318456757686519864962381827172729557652275340558205260377189870134584857824951376

$\frac{- 6+\sqrt{60}}{12}$:

2.230698266878966845929476908427025536292088558027152738966368240518809974357249384140799819233894570

$\frac{- 7+\sqrt{61}}{2}$:

1.632588184368224375753783701689005526555732732227631461014525861992624150675473301254323658658031098

$\frac{- 5+\sqrt{61}}{6}$:

1.799514861984147086581517739192482288676407373270322958804036720310802587012040090528836518772391887

$\frac{- 7+\sqrt{61}}{6}$:

2.222639647392633711318975160389740873954870826624279602122088770068724802765043848104298465354610834

$\frac{- 7+\sqrt{65}}{2}$:

1.754897874378163771190986554760949545654863917813581712910849438172630630518098186239953896260493185

$\frac{- 5+\sqrt{65}}{4}$:

1.582496050885300005746589570781502656314992512385751042868726301519514138799479879280006074053471486

$\frac{- 7+\sqrt{65}}{4}$:

1.746152439827328923870568218143744203223583395486402845410224269125149584861688436366806549204984888

$\frac{- 1+\sqrt{65}}{8}$:

2.112758960495885014171588096847792076139299113921763646738067530051313634024415608065915630343551150

$\frac{- 7+\sqrt{65}}{8}$:

2.252064180527440186506653609173158808985150473117600096867834803618369767112341856525844425339503472

$\frac{- 5+\sqrt{65}}{10}$:

1.718149506705533135271565722272617345749404455908906460047012598675434535077357360054691039207794427

$\frac{- 8+\sqrt{68}}{2}$:

2.388785477996206608109852654696204328030781587451165434755686900528538455174240241223395266460350637

$\frac{- 7+\sqrt{69}}{2}$:

1.569120273945457551229029368633986472013849922316138406636641185940367022615905381880430648553526251

$\frac{- 3+\sqrt{69}}{6}$:

2.104087811922881893537545099256980691222037090797705843422845843766552143918695232732328596585355029

$\frac{- 3+\sqrt{69}}{10}$:

1.750189526553979461504747868332369400160417380339730738077518557271025532245567849679674358352748951

$\frac{- 7+\sqrt{69}}{10}$:

2.240163485739767491020338167411248949752564409748284383481339741232805428878759766695803890737029604

$\frac{- 8+\sqrt{72}}{2}$:

1.986456441269314545816433519481662137516697063036199328517509404612904499435934957849515994207405329

$\frac{- 8+\sqrt{72}}{4}$:

2.350318342240116895593879372555606755767548535120515763132740348065729872188785244821551636189954841

$\frac{- 7+\sqrt{73}}{2}$:

1.542447467153873250677117955418936745329032150287563573733731450404535310193069149098479231207789362

$\frac{- 5+\sqrt{73}}{4}$:

2.113644364962109841589701981289132239134560288831228511898563993102037984366812541745513527630555793

$\frac{- 7+\sqrt{73}}{4}$:

1.500483461919831643184164103954344552825226175705345815731571532415908712249076314785053795344253749

$\frac{- 5+\sqrt{73}}{6}$:

1.421157117751829250578202259811745265506315884773173656190401612382154734071065881023498181053025934

$\frac{- 7+\sqrt{73}}{6}$:

1.901293061199755076487911603674898359218882480557725745248819934390997643229217134619478448957050471

$\frac{- 3+\sqrt{73}}{8}$:

1.514650890400339550876908463236829082247690291934886857208860539590414442155532492229996840183800080

$\frac{- 5+\sqrt{73}}{8}$:

1.656458168362158129522866489979394073335121398177670608180423321393332995322467964801706209917972665

$\frac{- 5+\sqrt{73}}{12}$:

1.662792810976459970978568756052924645649056543441407095615005465801124930198373915041246585026265950

$\frac{- 7+\sqrt{73}}{12}$:

2.248989827984661656867778853071832531107351567111979062504132379366826579793143230803844606634322180

$\frac{- 8+\sqrt{76}}{2}$:

1.604353316592574006936818984814138337617339613510493758259979386349534363921325485794367559297607230

$\frac{- 6+\sqrt{76}}{4}$:

1.639061942476995460610018529853647235622392604900941611536402227384514323702391307106081250546302353

$\frac{- 4+\sqrt{76}}{6}$:

1.615047641092852603572793311286141846249568283938874463629548114822249996683155008009482033352704167

$\frac{- 8+\sqrt{76}}{6}$:

2.315260283234952182068089491879601668564662825473833810065267294518302513783450757762039192539578447

$\frac{- 4+\sqrt{76}}{10}$:

1.843535958840685284501671665182269240856955322258585883073630971632539845581033604052488852676714330

$\frac{- 6+\sqrt{76}}{10}$:

1.748227049333168263972403282023198377862744795265348272674064030575153938606803569789867143249399838

$\frac{- 7+\sqrt{77}}{2}$:

2.199777495662190422220170710051755744352165364372366814008328458374295549458294252520711167147975472

$\frac{- 7+\sqrt{77}}{14}$:

2.344171735686426463454241117474100771855284040502225127181541299117593302221883770351888192235199944

$\frac{- 8+\sqrt{80}}{2}$:

1.863171185950228301861424294660710260884072359604118168572280411879471933891987533146169560411435758

$\frac{- 8+\sqrt{80}}{8}$:

2.356700182540333141556987077394672360270403880437158790409211054748454086915973990570163646800073023

$\frac{- 8+\sqrt{84}}{2}$:

1.417409911224853765586446643411035059355693125389333113304536320971800150602902743854374398951787824

$\frac{- 6+\sqrt{84}}{6}$:

1.791422555681224929463082832339321461089176459060525296172933471227684106067998743296932030974283524

$\frac{- 2+\sqrt{84}}{8}$:

2.183523942189162689279078279234733786683610836663957265372893562018333208597523585907655909468677293

$\frac{- 6+\sqrt{84}}{8}$:

1.636006103716291386286502368808063294302936629562810966487873176878764018156702607348771182355149442

$\frac{- 2+\sqrt{84}}{10}$:

1.505579043644340207906773815519823975194898071277640907320465646359679750293933838373633516907498639

$\frac{- 8+\sqrt{84}}{10}$:

2.314883778810726605681614012334221655610411685469873370601673329540212862206255420511453770276294331

$\frac{- 9+\sqrt{85}}{2}$:

2.481778011998106847067885740415758748203498707314526174655634853616456562041476250719400627038434549

$\frac{- 5+\sqrt{85}}{6}$:

1.408562453113636269801998285845468124103391734459463352836606259179845650563034649210827465339022951

$\frac{- 7+\sqrt{85}}{6}$:

1.515518658077640261881140314051416770310819377288761440879061201035716927587432743264735203165346676

$\frac{- 5+\sqrt{85}}{10}$:

1.502337754643325478884666128313997659854460414631814298737768333135383867323618092851339322552334657

$\frac{- 8+\sqrt{88}}{2}$:

1.509000969469146023995047273371331191743213793510360579439674708117615446093859524512335291020475347

$\frac{- 8+\sqrt{88}}{4}$:

1.824352928728166015134943093404004696922279985063663265123612091146062955352176597572018940339420770

$\frac{- 4+\sqrt{88}}{6}$:

2.203728477982713819940824122768240962903617124987520520008458945358775851970187683219307699660611635

$\frac{- 8+\sqrt{88}}{6}$:

1.888927605542166330001567375307575465147031191112266229876076787544409248089395096424511673502985068

$\frac{- 4+\sqrt{88}}{12}$:

1.649063829854613846309128513882183850624515462468050702425696226333176167981320959707976116765116515

$\frac{- 8+\sqrt{88}}{12}$:

2.335860456762515097483197543327603231146588719828382223242699592137054365970748764584197621393434265

$\frac{- 9+\sqrt{89}}{2}$:

1.830125241822575652093828730994470110117773735650430529925174749768537412356018322891865878658104971

$\frac{- 7+\sqrt{89}}{4}$:

1.392797145256907062962347889282154693630122219259891676233306561448489839585627511633610278677200039

$\frac{- 9+\sqrt{89}}{4}$:

2.419608537533503466233580406519000337686422524396795008891976270059845580308818029398103610910170315

$\frac{- 3+\sqrt{89}}{8}$:

1.827939762412725050593015410312101596903297687221643666592889379407311195803364955526899897555762803

$\frac{- 5+\sqrt{89}}{8}$:

1.603274987916215755736227252417360827465202435539853241406552423424072224496648880582252452694124487

$\frac{- 3+\sqrt{89}}{10}$:

1.472535698893684701562871495643992415635281852896354976682581941090191796769538503710189770126350458

$\frac{- 7+\sqrt{89}}{10}$:

2.001985003016270917519580269011622837008553124653107820659223792395528450479444353787806161668068157

$\frac{- 8+\sqrt{92}}{2}$:

1.768496613951038110058894322469702522053898261807933759805198697701939034410379849587384566262107461

$\frac{- 6+\sqrt{92}}{4}$:

2.240533064991610387565032697264018396272974236576141630902568028457011073956886303250277763055608487

$\frac{- 6+\sqrt{92}}{14}$:

1.935244834070803111776911725226326007067935599833241539730566099524857724231287077100461093593825378

$\frac{- 8+\sqrt{92}}{14}$:

2.375338683618147458494913892870156992856593035189375689306843383799837175711590415984261056595229619

$\frac{- 9+\sqrt{93}}{2}$:

1.918406587376347903539436830353779268025291969156620617398681323468763976149533851581729972991732001

$\frac{- 9+\sqrt{93}}{6}$:

2.438155775773056704685513828986008679397526005058102877891310695027575949340792488084423017051546281

$\frac{- 8+\sqrt{96}}{2}$:

2.304398779285770252476093873263245200957203013699748195843672888334047550662403700493917518622417564

$\frac{- 6+\sqrt{96}}{6}$:

1.310359835046859277045245800400521860042088113712661004405288711193490625904682084232181807792335467

$\frac{- 4+\sqrt{96}}{8}$:

1.384259787549754232873185389748219524431000655770981876259087915974136407782877818290550048104082637

$\frac{- 4+\sqrt{96}}{10}$:

1.347667284262623439056389448350736392702429825315412203882596821169168318669166813297626774449078426

$\frac{- 6+\sqrt{96}}{10}$:

1.465790572511121672103131798838228903707568593774644674053182842071742282280696527043293646432006842

$\frac{- 8+\sqrt{96}}{16}$:

2.444887509263264425152723543357488218220161245971435985340054154017825083017231703088589524877342372

$\frac{- 9+\sqrt{97}}{2}$:

1.557436995901221729927697779411904661158153560626904858637526261711144950033672979992564171124112365

$\frac{- 7+\sqrt{97}}{4}$:

1.528227284575584679299408470497088811768873969756347818841103982849626086252633340917271672009902179

$\frac{- 9+\sqrt{97}}{4}$:

1.874469870226477461557480600444502629835745105364439899625166728464765872435809027128151303217073449

$\frac{- 5+\sqrt{97}}{6}$:

1.734746823232823592865344296745767904439882615310745052188563408586576908287998928387524582753804950

$\frac{- 7+\sqrt{97}}{6}$:

1.888449034047901343196897822251140006793623973371579356509328273066855338311801587947157759830704020

$\frac{- 7+\sqrt{97}}{8}$:

1.650279276717649584674966355459906074237664246561409798854507694311315653841919815974998343428403150

$\frac{- 9+\sqrt{97}}{8}$:

2.408560455526087529039242545875335191981243549498319799463927454657248174020929748369544407028470117

$\frac{- 5+\sqrt{97}}{12}$:

1.669231880028313518898389794018589932423401833260392003081494993322478988859209865426922240279816514

$\frac{- 7+\sqrt{97}}{12}$:

1.882996683306203613683416985633841432857767486216826422290183278886549942314829604269998053068517062

Definition

The Brjuno function is $B(x)=\sum_{j\geq0}\beta_{j-1}\log(1/x_j)$, where $x_0=x$, $x_{j+1}=\{1/x_j\}$, $\beta_{-1}=1$, and $\beta_j=\prod_{r=0}^j x_r$, using natural real logarithms [1]. This table gives $B(x)$ at quadratic irrational arguments $0<x<1$.

Parameters

$x$

— Argument ($x$ is a quadratic irrational with $0<x<1$.)

Formulas

(1)

$B(x)=\log(1/x)+xB(T(x))$, where $T(x)=\{1/x\}$.

(2)

For a periodic argument with least period $\ell$, let $P=\prod_{j=0}^{\ell-1}x_j$. Then $\displaystyle B(x)=\frac{\sum_{j=0}^{\ell-1}\beta_{j-1}\log(1/x_j)}{1-P}$, obtained by iterating (1).

(3)

For an integer $a\geq1$, set $x_a=(\sqrt{a^2+4}-a)/2=[0;\overline a]$. Then $\displaystyle B(x_a)=\frac{\log(1/x_a)}{1-x_a}$.

(4)

$B(1/\varphi)=\varphi^2\log\varphi$, where $\varphi=(1+\sqrt5)/2$ is the golden ratio. The factor $\log\varphi$ is the regulator of $\mathbb Q(\sqrt5)$.

Comments

(5)

The fractional part is $\{u\}=u-\lfloor u\rfloor$, and the logarithm is natural. This is Yoccoz’s function $\Phi$, for ordinary regular continued fractions. The series $\sum_{j\geq0}\log(q_{j+1})/q_j$, where $q_j$ are convergent denominators, is a different numerical expression and does not define these values [1].

(6)

The current arguments are $x=[0;\overline{a_1,\ldots,a_\ell}]$. By Galois’s criterion applied to $1/x$, these are exactly the quadratic irrationals with $0<x<1$ and conjugate $x^\prime<-1$ [2]. The argument identified by $\mathrm{quad\_A\_B\_mK}$ is the positive root $(-B+\sqrt D)/(2A)$ of the primitive polynomial $At^2+Bt-K$, where $D=B^2+4AK$. Here $t$ is an auxiliary variable, and $D$ is the primitive polynomial discriminant, which need not be a fundamental field discriminant.

(7)

Quadratic irrationals have eventually periodic regular continued fractions, so their partial quotients are bounded. If they are at most $M$, then $\log(1/x_j)<\log(M+1)$ and $x_jx_{j+1}<1/2$; these inequalities give absolute convergence of the defining series.

Programs

(P1)

Sage

# B([0; overline(a_1,...,a_l)]) for a tuple of positive integers.
import numberdb.sage as numberdb
from sage.rings.real_arb import RealBallField

def brjuno_period(period, bits=429):
    if not period or any(a < 1 or int(a) != a for a in period):
        raise ValueError("Positive integer partial quotients required")
    F = RealBallField(bits)
    xs = []
    for j in range(len(period)):
        p, q, r, s = 1, 0, 0, 1
        rotated = period[j:] + period[:j]
        for a in rotated:
            p, q, r, s = q, p+a*q, s, r+a*s
        xs.append((F(p-s) + F((s-p)**2+4*r*q).sqrt()) / (2*r))
    P, S = F(1), F(0)
    for x in xs:
        S += P*(-x.log())
        P *= x
    return S/(1-P)

brjuno_period((1,))

References

[1]

X. Buff and A. Chéritat, The Brjuno function continuously estimates the size of quadratic Siegel disks, Annals of Mathematics 164 (2006), 265–312, Definition 1. (arXiv) (doi)

[2]

M. Waldschmidt, Arithmétique modulaire, fractions continues, corps finis, CIMPA–Mauritanie lecture notes (2016), §3.6, Lemma 47, pp. 37–38. https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/NouakchottCoursCIMPA2016.pdf

Similar tables

Regulators of real quadratic fields — The $D=5$ regulator is $\log\varphi$, the logarithmic factor in (4).

Markov quadratic irrationals — The reciprocals of $\xi_{1,1,1}$ and $\xi_{1,1,2}$ are the fixed-point arguments $[0;\overline1]$ and $[0;\overline2]$.

Eigenvalues of the Gauss-Kuzmin-Wirsing operator — The transfer operator there and the orbit sums here use the same regular-continued-fraction Gauss map.

Values of the Wilton function — The Wilton function uses the same quadratic arguments, Gauss-map iterates and weights, with a factor $(-1)^j$ in the $j$th summand. Absolute convergence at these arguments gives $|W(x)|\leq B(x)$.

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 all 141 periodic quadratic arguments with primitive polynomial discriminant $D\leq100$, across 40 discriminants and 45 complete Gauss-map cycles; the noncanonical discriminant bound keeps the radicals small while covering fixed points, longer periods and nonfundamental orders, and every phase is retained because the starting argument affects $B(x)$. General preperiodic quadratic arguments are not yet included)

How they were obtained:

Each least period is determined by exact integer transformations of its primitive quadratic polynomial. Arb ball arithmetic encloses the positive roots, natural logarithms, weighted cycle sum and division by $1-P$, using 429 working bits. Every resulting ball lies inside its written 100-significant-digit last-place interval.

An independent calculation uses integer convergents and a separate complete-quotient recurrence, with 4096-bit ball arithmetic and 800 terms. Its remaining positive tail is bounded by $4\log(M+1)2^{-400}$, where $M\leq9$ is verified from the exact period. This enclosure also lies inside each written interval. Constant-weight and logarithmic fixed-point controls precede the full check. The functional equation is checked for all 141 entries, with additional fixed-point and regulator controls. The attached checks repeat the comparisons on the stored values.