Values of the Wilton function

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Numbers

$x$

$W(x)$

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

0.2974052636752033248629120844760725702131211859825318086157485455706195855223040644958581163101632068

comment: The golden-section argument is $x=1/\varphi=[0;\overline1]$, and $W(x)=\log(\varphi)/\varphi$.

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

0.6232252401402305133940200802505680026506953123465672528987147760961700045470141804676690732356266281

comment: The argument is $x=\sqrt2-1=[0;\overline2]$, and $W(x)=\log(1+\sqrt2)/\sqrt2$.

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

-0.5789818958886893178571408497700033308554644479822278287989191747854703942960541242836859134451643676

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

1.216974620968918334643212614899075814132285517946049535857487113572089626431885580906352875531265270

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

0.9170905432090176062084930298531028708282200757904396627824859879402855077303279801300883486599661842

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

0.8859345757889776734307455893486633876281777538984119707121339591595613137650884588524075889734942551

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

-0.5500553397053062853496973163953137801778483181814851352607627555098431424849110289011870572090690180

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

1.021447106637442456554361491477002929021505841073866880463420122088145854991051308998797505562449289

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

1.167925633102210158541006496850661490022458280552288492415150071616175179695788758361801318149330023

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

-1.036867181577948549154101614259678654966036441195887085005237955962766805344680565272183384664357371

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

1.606192562897364564020358006788214781032297966245805118595299908571719407367814203528582416058052500

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

0.1431052786701217353890316184569822088503206198946827368127950302029624725251705467635622114311128414

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

1.460627247610387477639276519178857926573010721618906159442751581472768214831913165494214665681589679

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

-0.3510805792580815260797648503543211785346279253106173220874639037914044462801885182636593099851425324

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

-1.131133159354076988384630682281502301251967691646611750578710621291204483979749596231840319010645578

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

1.220936577997903231428810618104926489830224499487961879407951019833521937262185867255204854450218014

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

1.611523353493962965045950403271987388736304385636111727349365727596029769415164097248948789028908973

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

1.381230463776673674434886192499585322027307220994019301033916012169098284144323626702690223041496033

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

-1.347311395943732559205083626244209694622365402874160694180076166003254569775643883929486130245773001

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

1.853558094049348847622047364406765196610090844731694906698671808912205066336894686202000058492148231

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

0.8701453932391644819810799281562596400479804303657853551510370111688330242919293885097082550369971874

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

0.3168573456670583455512612916689005997621214254073949697176083755115920010726263485738094110094680470

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

1.519283206515793261991685650600047918976370355210888299094723798179748963555692124822265609706974315

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

0.08717641124976094627842571496879249615655028894342836249624283806350703424440800177747107812249427533

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

1.209019464713957723474579973628605969617574034980986233580029820999991651488844219731695289234521976

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

-1.099757225060646566768874603844678738589306663945209139250350995720614175509511406619671745014793530

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

1.494063633056279579394224396483413336338236030390880401163762076238678537983475133044036919213637936

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

1.564554255450998019763051858043921441085745723963750346067644764211623119489547829306186036343873046

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

0.6449823934918624743719588805646010473354095313891302973818985355432833066886249665595736551298234826

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

-0.1758904890828449954927629384456752825812972980219411158050240613868259870687348605595013311155849471

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

0.6983478328323120976289025282419386868714370833847006383145115913205443338917014556328340354950709390

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

0.2023891051831741457857703675175222293305445205933999145604526402342359928355051097241516069767958283

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

-1.289188039713843686889316398458406046096734214367861260138574156209532391647566458138546173361871719

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

1.499815758932017728680895338423083439826848700476260106919545029247588701950147745896428729242251478

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

0.2167402185060130512884444260091740659252087050767826147106304495429549551806048173645475286362994784

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

1.705243060849846816955322296802667737012800852999549944112985450119985066153986326673353301236025037

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

0.5963312429531242133386622743516338108274621328661487786168163630525093350442403508812732066339017322

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

1.748778385951959119834785026260605986018619207298476892490861934005436078266690152921194721505576838

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

-1.582498734604373038528175868876903604151701163066108432988714765112816499526657448245721201779844942

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

2.037465662498903626357068269068497455361098935225306100630266652964763003783701516713334672985143573

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

-0.1061021983318904867597222498531815557218858989287987052248566859179606561683461676339509004972702487

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

1.882678108464037136131052002562895068026220258341865477233039723138641914413880837903338008775323522

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

0.7832756696407148815505709943197058825680489285751058440831550344212718444568232532805788189093153953

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

-1.395727071614961137281755480366777523466129188123825247393164726592784113206724410102757345642876868

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

1.372115577260756547762304700750027113629906643826603439415315579666930420899818030109910458816131195

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

-0.4651292868597062698559914382206216495589479138265179359932808103301026592484502247001358521632310013

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

1.769332002649343310352037394325539260476104766921650460046605080757433435103732425117737073512061277

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

1.724233100247575021252548544956605062190466636115332238209170336079996898574191789565853292418884937

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

-0.9071405390156978268208218082820364994791413905690650016139266157741628246957382915303981902936942966

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

-1.640673973607954232931209892262447135990520924747680172927247455766708766047333837446425011099693162

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

1.626099120518011643734425280855426832992172528446161056644994568032808023470672333233697316723464592

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

2.042863294097812849597961655220457575703637833626479817209989727502962411604738336644782176644151802

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

1.281684498559821999850751413039302729173565191885760610770150500516451129500914188275863273212844908

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

0.03492234839430274341220070674224604589759294261577775387625263093334078087730746812795272827046421944

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

1.808253865858393786735124960529331417551981514251983353139085452033725063390889552472957679624010338

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

-0.1533435188850362741128074256722052589122435682678617922445801973228406020294574491021632194515090417

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

0.6515616500627044026510873173176782150099777024213736099622370418357593518646677302832105383953319985

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

0.5670696151025280442378361554916043699992834554731225101925884631649042643059314851612793848757970132

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

-1.771995300890776834108954400525828033052610188199749631674360254825122404572678220658382420059589672

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

0.3323755953299546233942658020333191142793755866131845456615392744904259278626474740260828296159209999

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

1.137731635365587107068700738634592286667514818365763456911602551017604536511024649963717998296195734

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

2.185418666934382588277009606053787269836488431743754773302925182246307486414065012590654165134989645

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

0.9518021296888501308614389019514369299034861062349792438627614468046853034709536971429533292927112494

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

-0.1190796210379850353219323508453163728289147888152784816647642992403653879751971090079850758482148536

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

1.873639411662460147181990124611540382323058352649197286614070616350746642084964160552307886238791921

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

1.424515891928537496754399621418917959992615875657218928397274292489459564933665813115760061991694451

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

-0.3087898863650568763241209890949852961295957348808541422065850536174250933845208099298478012672440677

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

1.407901379248015048943689497928710246866349708885962898187999544432207590165444178319002465722534937

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

-1.490721638614620164440915058224080123702031261604003656956508256771055830386993051449689954506832956

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

1.829894566989144231399541028341885041983476649194203101022555133233600428855826824172250788961640372

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

0.7522969321041227404969562402787363623043574525410990303761596930996456082886511745364643793712777447

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

1.865107341180931284548646943179822164667493825401497118283985405546976467750333428935787925088165437

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

-0.5934818309024363028880472699216237400444909477920213478260641836738991091643097048709397524927756325

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

-1.745733122360947336958591102298697006900032061614137326759269140533191782046211848597776750253096989

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

1.560024171852183394143076000862575350716151788994256684644804756026538172009385085505272362637693878

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

2.112684238613742633477576519444698709518202093672096602801907736341967295314266921775337360614201672

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

0.9317960591097851932295161641373256864696617011845001042340437935861789097214851911143432705480446146

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

1.996274946300338546654728725247982672262905887784729964521177544925715482823984040355431106109379155

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

-0.4935144617962278356600980195487313373488653181651274886899490856589722712431855944971971519009949285

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

-1.751992441353247618300830489779969005227113905779868087164331657815547582100535286976882975281327837

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

0.8302216411501684188353611047293016475912082497119881292571131962085340467874230543576925563383056627

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

0.3152682576460029094876873405652567845818779660883267171856890505510872600961668769717057962993197297

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

1.808227740440859585552312609189556924777510292316999989312430664379847381715179944368098093754906746

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

0.3576194674969413542899953374827964436837025638318960376776361056152142689264715061469108720098955826

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

0.01313871656870450717766881033343588229581938151901588791682819641750183415308539667948742293081045859

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

0.9744571764338964101210973221254922376442791734530687174613294329306377645348882255384479097559703311

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

2.114026787246380114586221426473216923017047355504831720110459684938886948498894693872433699038083307

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

1.267103063310229025598264113376703840241811719981704676754968984774518724052041594820712108928736845

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

0.5080828873854298662136023036988504632763611266235365261114165284451705963891352977179717379083391125

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

-0.6753673791359799244037305132742850932601847275322983694434237958451429654853360180147073123739013777

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

1.971739396178204524691546393816038371288210985386754595049548056672740696885754222408201691933874544

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

-0.1789838441046950276934844241860377715549772447541146275036219276932332155589257724523784296040662006

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

1.164676481147394457099196139553104277673472380888108622388625915956063707052940724423912834297627553

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

-1.930776444998231609561460166463551756748922720798649344336793350562166245204956294421231534712632641

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

2.310066929607884260372680347527108194451438522501183158406349654965978720913439197299513091565933507

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

-0.2736114697885273500490845670705715765969807779131186445314541041344132909023725056957220539531533548

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

2.169078108885616883287975922690940184937890663237977214193617430491679591702231338662622406707135089

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

0.4736848960500469752489508504574597037932113002462710525655242030930732334639035426592988788227208808

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

1.571037317839863813445174255403550187496031684856991516287159741824788211150227662033410831082619990

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

-1.765752005193420975933002628979302587647213199460924940519587717110936948150979822790607755035093021

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

0.3056692859416006591359862846180219197093359083253406089763837572857961641033572678391029213822649288

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

0.1143392319478854010169737264361985043049484897308755717111075740924899166756287840923099524706014258

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

2.094542604495615481498102252930736795590228845663603797132599433996933535874316066471431308090995131

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

1.990811854528032563133290966881134971081727689231878082676928367708567439228462969635189769891205086

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

0.03633084534045147880902842498001271334895307634612351851856879232900207462067081989051913285352610914

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

0.9810178394170722547401828475907277838190801277729461507022490968610892965747651818893245073960199486

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

0.4489130806000042343454910852784201168736101107953746727084554739767833547913897784393241774564892304

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

0.1510968294255752862289490703846599576186771975435687036402225255476337697990331054458606568270313194

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

1.690016500574909103651147801627807811919333221220467884032991721605881391609523521729102808739884868

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

-1.814582007202229914630618526714284866438412234405175104030523136471129806714085701159822250102875323

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

1.080135592264661295694629722253973614477308576606005179902326022511583231886654910331394882160955661

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

0.3177280933060742349063146103275634022828809996058563047321333625826366831538105680124500612281468212

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

2.144834176228331436554691166060386855579418797061426467799341758947734098744359862536853155373041232

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

1.404016168668305171751818976093589501973981820739220844786308221479197252506980951652701591047781440

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

0.5709245084240802263946597420230997579804661027462113257398927099236121364016167028239311054459175252

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

2.068719449894509999785233649882802938482787656125108112875554703413465691457886881449852716619508276

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

-0.5136428823751825617069520543297584690042750744220189733522116518053891141821062491870899915365383344

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

0.8748290419153001232379644674187630792969753802509717525255947301456827274512136411632074001376424063

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

-0.1218717582272561421587181344587242811884179965756827798863973964841310929346692418567751157471570615

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

0.9095344394281334720118471522138937697296669513111957617720873673497806881596302973364808967449242842

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

-1.257133729639427599660811876949405886830340983061022298222167971584676063238812412575001966624853579

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

-1.972972931031839565265547561509253106140639629363715520781236934648979985501799055289952401720353926

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

1.866602987935555891922240952947316578116151054107071652014263671499873599599571114933489018169790170

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

2.317201726019962463374230481388981784783912396755387125537118885448583428707734116842814991122593301

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

0.4301676101786083380793012076421989992279454233618951141894775756008990906260364130434390570109584931

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

2.186212180855612336508856451842709857497852835976027588318174853213286385992650890099129546127673354

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

-2.067474431955243363576389802459299934891165629396727814123383584977481170970537019807727538451857755

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

-0.03013435498104132970109055020213924820802321532965482518430935374373738673067938983990536926240505353

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

-0.3879999832101175154730557067996786505370575031774015436743373305643019407405471014250447156448178493

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

0.7700399372476316931559270847852015879201576700957726976852664175611521080793387975379510705106858102

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

0.9795661878022559990693622517247257380503678533015960184665990839906082190507480085898780011025288697

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

2.418263743278096705209087174985802502295740266354567924684682101973929209290385149755281387990371973

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

0.3227038967318959530547742156613765207608899176870754484900978327980272730081369215997345944613113347

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

-0.3935058317709087730923063692783486940482658801614934974228547520731189983427479278120184241299908744

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

1.633665580706812566881189045228451129433253626678894490682090914871032064032021366137330328212233318

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

-0.6356537705382743972018400455068898671333566329366012371374086216078396127036401285038809953215765024

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

-0.3040435050779223577062322764902110671669031609210970703451301646664016637648859377211221916202098124

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

1.258884299459054120283361652366113470351865175647998500680879864230123963276688005849375414130014823

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

2.209063922377648226748732795451690210102301312036332591085648551025745018479588787440767548700165263

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

1.029018787498732950450366171231864316010164754904724630171355069950020067840582819247932401160249029

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

1.050148429210470344922893432600244905994332460365696387296892552148832601366026328827725985435336491

Definition

The Wilton function is $W(x)=\sum_{j\geq0}(-1)^j\beta_{j-1}\log(1/x_j)$, 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 $W(x)$ at quadratic irrational arguments $0<x<1$.

Parameters

$x$

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

Formulas

(1)

$W(x)=\log(1/x)-xW(T(x))$, where $T(x)=\{1/x\}$ [1], Proposition 8.

(2)

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

(3)

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

(4)

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

Comments

(5)

The fractional part is $\{u\}=u-\lfloor u\rfloor$, and the logarithm is natural. Ordinary regular continued fractions are used. The series $\sum_{j\geq0}(-1)^j\log(q_{j+1})/q_j$, where $q_j$ are convergent denominators, has the same convergence set but generally a different sum [1], Proposition 7.

(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.

(8)

The Brjuno function $B(x)$ uses the same iterates and weights with every sign positive. At quadratic irrational arguments, absolute convergence gives $|W(x)|\leq B(x)$ [1], Proposition 6.

(9)

The multiplicative fractional-part autocorrelation is differentiable at precisely those positive irrational arguments for which the Wilton series converges. In particular, it is differentiable at every quadratic irrational argument here [1], Theorem 1 and Proposition 7. The theorem concerns differentiability, and does not identify $W(x)$ with the derivative.

Programs

(P1)

Sage

# W([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 wilton_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 j, x in enumerate(xs):
        S += (-1)**j*P*(-x.log())
        P *= x
    return S/(1-(-1)**len(period)*P)

wilton_period((1,))

References

[1]

M. Balazard and B. Martin, Sur l’autocorrélation multiplicative de la fonction « partie fractionnaire » et une fonction définie par J. R. Wilton, §§2.2 and 3.1–3.2. (arXiv)

[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

Links

[3]

Balazard and Martin, full text on arXiv

Similar tables

Values of the Brjuno function — The same quadratic arguments and Gauss-map weights occur with every sign positive.

Regulators of real quadratic fields — The $D=5$ regulator is $\log\varphi$, the numerator 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.

Multiplicative fractional-part autocorrelation — Convergence of the Wilton series characterizes its irrational points of differentiability [1], Theorem 1 and Proposition 7.

Data properties

Entries are of type: real number

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 $W(x)$. 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, signed cycle sum and division by $1-(-1)^\ell P$, 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 absolute remaining 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. A nonzero alternating geometric-sum control and logarithmic fixed-point controls precede the full check. The functional equation is checked for all 141 entries, with additional fixed-point, Brjuno-bound and regulator controls. The attached checks repeat the comparisons on the stored values.