Multiplicative fractional-part autocorrelation

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Numbers

$\lambda$

$A(\lambda)$

1.260661401507812622954147382728832848680635611335643226828535846080663665076856124452539533980161646

comment: The squared norm of the fractional-part function in $L^2((0,\infty),t^{-2}dt)$ is $A(1)=\log(2\pi)-\gamma$.

1/2:

0.7722092559908731398613025066820804944916016749116686065912318821871493433152184144379388187360165624

1.544418511981746279722605013364160988983203349823337213182463764374298686630436828875877637472033125

1/3:

0.5750034694621839543151319682696108715870416798435534770613416159183956334853874208818280481717559658

2/3:

0.8822070185588043526546718409387383104560831473085791796675616546327516545155414016278172632243569063

1.725010408386551862945395904808832614761125039530660431184024847755186900456162262645484144515267897

3/2:

1.323310527838206528982007761408107465684124720962868769501342481949127481773312102441725894836535359

1/4:

0.4644025313717859846873332345668570346310952437755296899882589336996080995886521034926918900741693297

3/4:

0.9463438999232604288591479667519545389071304522660544349161134899405190285895282335929612610935401362

1.857610125487143938749332938267428138524380975102118759953035734798432398354608413970767560296677319

4/3:

1.261791866564347238478863955669272718542840603021405913221484653254025371452704311457281681458053515

1/5:

0.3924788623380489287209395467692375847959430290010881051753705914945552607991954860718586035482807417

2/5:

0.6328104682494277693777625189711642154862935071434277411256695803440756946651627503003714382806477414

3/5:

0.8206693108890392471708282970762108247145385096619800188916236147690764816267109659970489064309416069

4/5:

0.9895033524980985422164804541665283262871682077089242159839924634034202876705092113444581650417615124

1.962394311690244643604697733846187923979715145005440525876852957472776303995977430359293017741403708

5/2:

1.582026170623569423444406297427910538715733767858569352814173950860189236662906875750928595701619354

5/3:

1.367782184815065411951380495127018041190897516103300031486039357948460802711184943328414844051569345

5/4:

1.236879190622623177770600567708160407858960259636155269979990579254275359588136514180572706302201891

1/6:

0.3415025817467434728891245129191679404843597553320172556821620046790966807409416822805560806643209471

5/6:

1.020943933942957579676550201771256608967066931691579423328432117529160322285949574588015197906490766

2.049015490480460837334747077515007642906158531992103534092972028074580084445650093683336483985925683

6/5:

1.225132720731549095611860242125507930760480318029895307994118541034992386743139489505618237487788920

1/7:

0.3032631804979138170879793942343695288261454981645807120197316661023338672098610245481058223459002672

2/7:

0.5005626616257090172294659004680328405423410711896592633822339761853964057971061742676158069288294754

3/7:

0.6622673700319948320978434041525089535867688074558288199496967631116529828109722863631781149961938519

4/7:

0.7977079219388736369033973508867787424041470366945070888990319363133481854705356680677781674872368057

5/7:

0.9129714229104948235731398318791339639370337547477231212924136006263658371272371561029595082763401814

6/7:

1.045056560505760130423802924295318378046680692457434442244694113219146127921040629028000709399808188

2.122842263485396719615855759640586701783018487152064984138121662716337070469027171836740756421301870

7/2:

1.751969315689981560303130651638114941898193749163807421837818916648887420289871609936655324250903164

7/3:

1.545290530074654608228301276355854225035793884063600579882625780593856959892268668180748934991118988

7/4:

1.395988863393028864580945364051862799207257314215387405573305888548359324573437419118611793102664410

7/5:

1.278159992074692753002395764630787549511847256646812369809379040876912171978132018544143311586876254

7/6:

1.219232653923386818827770078344538107721127474533673515952143132089003815907880733866000827633109553

1/8:

0.2733960883474763496439985167048740182095005480753890468114301634143098827321124375970679811877215424

3/8:

0.6037387280011130037438694016869341817446691274398957178477701168037361320102617762485821368493408133

5/8:

0.8390593737333581187655432580834227872424577536947794499049681279102210147658788612790377640872247004

7/8:

1.064235515467792470119874479048158447796986741805943691804334066863496673152860314583369749982668599

2.187168706779810797151988133638992145676004384603112374491441307314479061856899500776543849501772339

8/3:

1.609969941336301343316985071165157817985784339839721914260720311476629685360698069996219031598242169

8/5:

1.342494997973372990024869212933476459587932405911647119847949004656353623625406178046460422539559521

8/7:

1.216269160534619965851285118912181083196556276349649933490667504986853340746126073809565428551621256

1/9:

0.2493512839652767500564796714511129348857892120949519505391898796338723641924997115470100629393841343

2/9:

0.4174743860789968194020750943993439889582256770860189139570965106281924847112244228841393650859059354

4/9:

0.6804800993925869264221509963356909618108230339415405814798722057014516701633837282508483421510545576

5/9:

0.7864245985377298812759988261073736269503047930887905715233362049279536345282489105193064179517562592

7/9:

0.9676604304977622230337412766053559890306620637851397786260987093230544407227649983321153830311653282

8/9:

1.079912814818776682631175095592650982767578729965294649436829568502097288276107011370458579944689488

2.244161555687490750508317043060016413972102908854567554852708916704851277732497403923090566454457209

9/2:

1.878634737355485687309337924797047950312015546887085112806934297826866181200509902978627142886576709

9/4:

1.531080223633320584449839741755304664074351826368466308329712462828266257867613388564408769839872755

9/5:

1.415564277367913786296797886993272528510548627559823028742005168870316542150848038934751552313161267

9/7:

1.244134839211408572471953069921171985896565510580894001090698340558212852357840712141291206754355422

9/8:

1.214901916671123767960071982541732355613526071210956480616433264564859449310620387791765902437775674

1/10:

0.2295322051740821208266533640334123075718977967563525625689560671215380245261282757929123351054623012

3/10:

0.5177285985609940850100116562129045178225269706459143904152757151642602086440698521231636301790216324

7/10:

0.9010878109417681888997906278239942556383993006214863560823746171345513355078062214223248194814914083

9/10:

1.093003865206755457425050708889513212888138863190835438289411620419569327382049419120191832307704796

10:

2.295322051740821208266533640334123075718977967563525625689560671215380245261282757929123351054623012

10/3:

1.725761995203313616700038854043015059408423235486381301384252383880867362146899507077212100596738775

10/7:

1.287268301345383126999700896891420365197713286602123365831963738763644765011151744889035456402130583

10/9:

1.214448739118617174916723009877236903209043181323150486988235133799521474868943799022435369230783107

1/11:

0.2128848370316622964047862582564087097497232619892527260250885813106400760554219336861106974318639311

2/11:

0.3599326720695457642660915537896074607006677980453488272049703803947905512443699088696590142730597609

3/11:

0.4833654852615073827591677879978701123746020758912110291133013886967679504735265333949087263351770034

4/11:

0.5920065982762652351762344086536526524597823318015489567324989436866880582643138348422299192524065371

5/11:

0.6930088049919366626072629573765724598068358902468873751994508176222509492015998997057371736529199150

6/11:

0.7800264867009780898612427835365244272161623897900277151396498523706780083322517688039898361534103308

7/11:

0.8485462302419524124616179618281870483882567932780297550670993311876470953836303854504827357676997797

8/11:

0.9234023183680665390776994899645236161983246965281193346361653799828683535145607115718707855502444184

9/11:

1.005229025155800123620094503898024720844291318006964952500354120928274903997521970265867657168799117

10/11:

1.104124156868878657096252145831308953904103050623808322817454335300486167396456303238820516774490728

11:

2.341733207348285260452648840820495807246955881881779986275974394417040836609641270547217671750503242

11/2:

1.979629696382501703463503545842841033853672889249418549627337092171348031844034498783124578501828685

11/3:

1.772340112625527070116948555992190412040207611601107106748771758554815818402930622447998663228982346

11/4:

1.628018145259729396734644623797544794264401412454259631014372095138392160226863045816132277944117977

11/5:

1.524619370982260657735978506228459411575038958543152225438791798768952088243519779352621782036423813

11/6:

1.430048558951793164745611769816961449896297714615050811089358062679576348609128242807314699614585606

11/7:

1.333429790380210933868256797158579647467260675151189615105441806152016864174276319993615727634956797

11/8:

1.269678187756091491231836798701219972272696457726164085124727397476443986082520978411322330131586075

11/9:

1.228613252968200151091226615875363547698578277564068275278210592245669327108082408102727136539643365

11/10:

1.214536572555766522805877360414439849294513355686189155099199768830534784136101933562702568451939801

1/12:

0.1986834979551244917197802474400802153245308789816865042614401800402613418551751346449531718254468345

5/12:

0.6478955883615461935992763458790797701821390956372857003378599767872238505270701025425419628614310948

7/12:

0.8057358435149588800219006101195249831078374542968834059556803978022605333701240679967310332341858724

11/12:

1.113704345970298247185623938671705459312387950627124613410576555947861201987052056803250038231846296

12:

2.384201975461493900637362969280962583894370547780238051137282160483136102262101615739438061905362013

12/5:

1.554949412067710864638263230109791448437133829529485680810863944289337241264968246102100710867434627

12/7:

1.381261446025643794323258188776328542470578493080371553066880681946732342920212687994396056972890067

12/11:

1.214950195603961724202498842187315046522605037047772305538810788306757674894965880149000041707468686

1/13:

0.1864112103177554873877635004235692507611095734062426309068821600700501499398337103078739699176656344

2/13:

0.3174800380064651686100422519499578826228318475203552243990032582581469933145741076505324970109367137

3/13:

0.4288296117186167326551223973795831848015525509315017315073910983282143605073096381252452525809634836

4/13:

0.5279397314041999754366187718055926341965767296910093200204854609785593486990399964080037543436328926

5/13:

0.6130954276977901051739995118791467439491734900986048974532541937610817140313092800655580630467278258

6/13:

0.7022272344746209038686346448733356193227218889374651475875026235549883768934614975999288135460315011

7/13:

0.7760640923730450805412911053236428898957054021171183189458833400176518899553770669906116161205515534

8/13:

0.8307709825924433497251756773924860495334179202071342291249531174407680059243042076931938878199406818

9/13:

0.8952023925102328024158229351526313944219497449045019882457444676183787641245889705404123079677266678

10/13:

0.9599824867186070459094306254077176517227577792807404726440403121850790151116009637599720234834769523

11/13:

1.033078267364116525446971324761759493255341607551714199379815724341110847113966591339520734327023835

12/13:

1.122055656627711518463918611392930113108070181475435330865485413465713898798670555541146887322618867

13:

2.423345734130821336040925505506400259894424454281154201789468080910651949217838234002361608929653247

13/2:

2.063620247042023595965274637674726237048407008882308958593521178677955456544731699728461230571088639

13/3:

1.858261650780672508172197055311527134140061054036507503198694759422262228865008431876062761184175095

13/4:

1.715804127063649920169011008368176061138874371495780290066577748180317883271879988326012201616806901

13/5:

1.594048112014254273452398730885781534267851074256372733378460903778812456481404128170450963921492347

13/6:

1.521492341361678625048708397225560508532564092697841153106255684369141483269166578133179096016401586

13/7:

1.441261885835655149576683481315336795520595746788934020899497631461353509917128838696850144223881456

13/8:

1.350002846712720443303410475762789830491804120336593122328048815841248009626994337501440067707403608

13/9:

1.293070122514780714600633128553800903053927409306502871910519786559880437068850735225040000397827409

13/10:

1.247977232734189159682259813030032947239585113064962614437252405840602719645081252887963630528520038

13/11:

1.220910679612137711891875201991170310210858263470207690176145856039494637498324153401251776931937259

13/12:

1.215560294680020811669245162342340955867076029931721608437609197921190057031893101836242461266170439

1/14:

0.1756890971640768001065552180892604403210351989242861452745743110413016815030817083993031436817854453

3/14:

0.4059987519668340843762003116267605858336383997647581505495186901339543538225627184514168891883516556

5/14:

0.5859446301043568389763875580968716929566471035641710801561783445190246189802754572658540985794146188

9/14:

0.8543971613056698403266884564908946556672790432200678192468863201157579336636176260693269690459122168

11/14:

0.9746598969669052127298262959491491812843652344707725665283020173982828422190905028448774831369532060

13/14:

1.129409107938307238471199294737535058869641272956191750157731019625530404557642738649303829027147873

14:

2.459647360297075201491773053249646164494492784940006033844040354578223541043143917590244011544996234

14/3:

1.894660842511892393755601454258216067223645865568871369231087220625120317838626019439945482878974393

14/5:

1.640644964292199149133885162671240740278611889979679024437299364653268933144771280344391476022360933

14/9:

1.329062250919930862730404265652502797704656289453438829939600942402290119032294084996730840738085671

14/11:

1.240476232503333907110688013026189867089192116599165084672384385779632708278842458166207705810667717

14/13:

1.216286731625869333738214625101960832628844447798975730939094944212109666446692180083865662029236171

1/15:

0.1662327771191725938884472609807079800065313954824739381385459274643841171978413455225658297811607290

2/15:

0.2847351500316032203977216305361040703704258289588845243493321829797870715931549201757460838123304574

4/15:

0.4761607572544388280474905978325642002449663120061860336272900859232167066873713566745003569323371971

7/15:

0.7093320272394836438177213121920344057489168217618285694250086843120703113965795768418262989786361205

8/15:

0.7734599945275759489273915115914094677126189735486561868125692607867718044499072846051132631397979012

11/15:

0.9286812306135845497052009438106517495743896022114138733270633688512968552802869331995499148461444662

13/15:

1.054738384134703121938120118250518771117434212535539744190967096289161611198422790602253373747827181

14/15:

1.135940101677794029611022310511044213153064048643190443993287570766778769688914199934425340454546916

15:

2.493491656787588908326708914710619700097970932237109072078188911965761757967620182838487446717410936

15/2:

2.135513625237024152982912229020780527778193717191633932619991372348403036948661901318095628592478430

15/4:

1.785602839704145605178089741872115750918623670023197626102337822212062650077642587529376338496264489

15/7:

1.519997201227464951037974240411502298033393189489632648767875752097293524421241950375342069239934544

15/8:

1.450237489739204904238859084233892751961160575403730350273567363975197133343576158634587368387121065

15/11:

1.266383496291251658688910377923616022146894912106473463627813684797222984473118545272113520244742454

15/13:

1.217005827847734371467061674904444735904731783694853550989577418795186474459718604541061585093646748

15/14:

1.217078680369065031726095332690404514092568623546275475707093825821548681809550928501170007629871696

1/16:

0.1578243892904763260357205240798967238970730445031494052480552994233204333860023050469188528829232320

3/16:

0.3680559848531748485464418965973067988120321157915498523046893095116370510369060049538227158375340777

5/16:

0.5347927290941974426734287173438938626978403868271961912298034036375841837442076105852318704807410842

7/16:

0.6716406974856947891388759948359763733612781061909971589456688682703506580930512042263629989185442322

9/16:

0.7906725925388799523389079714641323577204303706746053866949446407876800208634534967712239426726317320

11/16:

0.8917690385679490750414094564070844314385815729781146458372287436280649150188556595462837907209306518

13/16:

0.9997406935875078334455617051834817193817070168101449252776066174594611491025359952892306159083092615

15/16:

1.141784411413330953290009065159206099969822315169584117991199002689512948058628180013682132949354081

16:

2.525190228647621216571528385278347582353168712050390483968884790773126934176036880750701646126771712

16/3:

1.962965252550265858914356781852302926997504617554932545625009650728730938863498693087054484466848414

16/5:

1.711336733101431816554971895500460360633089237847027811935370891640269387981464353872741985538371469

16/7:

1.535178737110159518031716559625088853397207099865136363304385984617944361355545609660258283242386816

16/9:

1.405640164513564359713614171491790858169653992310409576346568250289208925979472883148842564751345301

16/11:

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16/13:

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16/15:

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2/17:

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3/17:

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4/17:

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5/17:

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6/17:

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9/17:

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10/17:

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11/17:

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12/17:

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13/17:

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14/17:

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15/17:

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16/17:

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17:

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17/2:

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17/3:

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17/4:

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17/5:

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17/6:

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17/7:

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17/8:

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17/9:

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17/10:

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17/11:

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17/12:

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17/13:

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17/14:

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17/15:

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17/16:

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1/18:

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5/18:

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11/18:

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13/18:

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17/18:

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18:

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18/5:

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18/7:

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18/11:

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18/13:

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18/17:

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1/19:

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2/19:

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3/19:

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4/19:

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6/19:

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7/19:

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8/19:

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9/19:

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10/19:

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11/19:

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12/19:

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13/19:

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14/19:

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15/19:

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16/19:

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17/19:

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18/19:

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19:

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19/2:

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19/3:

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19/4:

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19/5:

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19/6:

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19/7:

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19/8:

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19/9:

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19/10:

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19/11:

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19/12:

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19/13:

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19/14:

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19/15:

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19/16:

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19/17:

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19/18:

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1/20:

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3/20:

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7/20:

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9/20:

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11/20:

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13/20:

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17/20:

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19/20:

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20:

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20/3:

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20/7:

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20/9:

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20/11:

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20/13:

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20/17:

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20/19:

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1/21:

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2/21:

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4/21:

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5/21:

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8/21:

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10/21:

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11/21:

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13/21:

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16/21:

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17/21:

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19/21:

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20/21:

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21:

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21/2:

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21/4:

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21/5:

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21/8:

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21/10:

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21/11:

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21/13:

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21/16:

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21/17:

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21/19:

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21/20:

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1/22:

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3/22:

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5/22:

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7/22:

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9/22:

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13/22:

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15/22:

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17/22:

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19/22:

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21/22:

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22:

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22/3:

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22/5:

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22/7:

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22/9:

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22/13:

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22/15:

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22/17:

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22/19:

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22/21:

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1/23:

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2/23:

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3/23:

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4/23:

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5/23:

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6/23:

0.4700489571187882773187282547953726311692928641329524114769891896799156585203099534967120973792720436

7/23:

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8/23:

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9/23:

0.6204794362364674922989302318537325660331269776323157492835552697716306488956382693629011551121034392

10/23:

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11/23:

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12/23:

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13/23:

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14/23:

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15/23:

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17/23:

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18/23:

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19/23:

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20/23:

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21/23:

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22/23:

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23/2:

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23/3:

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23/4:

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23/5:

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23/6:

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23/7:

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23/8:

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23/9:

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23/10:

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23/11:

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23/12:

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23/13:

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23/14:

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23/15:

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23/16:

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23/17:

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23/18:

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23/19:

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23/20:

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23/21:

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23/22:

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1/24:

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5/24:

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7/24:

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11/24:

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13/24:

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17/24:

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19/24:

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23/24:

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24:

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24/5:

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24/7:

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24/11:

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24/13:

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24/17:

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24/19:

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24/23:

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1/25:

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2/25:

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3/25:

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4/25:

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6/25:

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7/25:

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8/25:

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9/25:

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11/25:

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12/25:

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13/25:

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14/25:

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16/25:

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17/25:

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18/25:

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19/25:

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21/25:

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22/25:

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23/25:

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24/25:

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25:

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25/2:

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25/3:

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25/4:

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25/6:

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25/7:

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25/8:

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25/9:

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25/11:

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25/12:

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25/13:

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25/14:

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25/16:

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25/17:

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25/18:

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25/19:

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25/21:

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25/22:

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25/23:

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25/24:

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1/26:

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3/26:

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5/26:

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7/26:

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9/26:

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11/26:

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15/26:

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17/26:

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19/26:

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21/26:

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23/26:

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25/26:

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26:

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26/3:

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26/5:

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26/7:

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26/9:

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26/11:

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26/15:

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26/17:

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26/19:

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26/21:

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26/23:

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26/25:

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1/27:

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2/27:

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4/27:

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5/27:

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7/27:

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8/27:

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10/27:

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11/27:

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13/27:

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14/27:

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16/27:

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17/27:

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19/27:

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20/27:

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22/27:

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23/27:

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25/27:

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26/27:

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27:

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27/2:

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27/4:

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27/5:

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27/7:

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27/8:

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27/10:

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27/11:

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27/13:

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27/14:

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27/16:

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27/17:

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27/19:

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27/20:

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27/22:

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27/23:

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27/25:

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27/26:

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1/28:

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3/28:

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5/28:

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9/28:

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11/28:

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13/28:

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15/28:

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17/28:

0.8244825246154645396943603602697165780339804545295329601769230913032311682180104096826648339929186274

19/28:

0.8861037704102971784635627072624255617045480122809205037012288731432449281541258436602452145240714093

23/28:

1.008232912520947874877744207021089146678631892107545918445656150304574431876351041593864494385701875

25/28:

1.084308967673164887530588940940374581052203406709342379539248690463906845339491817523064700897867928

27/28:

1.182980401475525261790424787306291896509820578396155923612268051989399505698745621561338672520724618

28:

2.801328182312126450669714410134488058498791262907956514382966285305508529597681199339015649655481753

28/3:

2.244536889373065651183507641085651271051807683391783888843892222709606427270427689907446850439110264

28/5:

1.985300356965648653301185988870805960370932465322927309315807844348688694173990380128110308782233912

28/9:

1.707717637061789364696117432162459035357604813249945556319551881077113094094620087770825991497108205

28/11:

1.584147883846403835994591306229733220832084119286855950539329784038417880857031356101671150346916999

28/13:

1.520342464030421754616098177032137326631826584254540371855226519139427040139037650172186214778429934

28/15:

1.445788477410501787062594468400251798340320381034315199014261701899492381279737917190030145905162630

28/17:

1.357971217013706300673064122797180246173614866283936640291402738617086630006134792418506785400101269

28/19:

1.305837135341490578788408200176206090933018123361356531770232023579518841490290716972992947719684182

28/23:

1.227413980460284369416384252025673743782682303435273292020798791675134090979905615853400254034767501

28/25:

1.214426043793944674034259613853219530778467815514463465083958533319575666780230835625832465005612079

28/27:

1.226794490419063234449329409058376781565739859077495031894203905766784672576476940878425290021492197

1/29:

0.09719672648835399627026570495754973986442038971015395311699006134576292677338971911715973210958332096

2/29:

0.1700845211083431691335601689533705120304519398269952071353919068497732926159430406235829921778528281

3/29:

0.2339932467294271061538321237607093803529867133296220903565982407848977178610838866182022339161628080

4/29:

0.2922422966952260226308126660835423348578691758659569391298745172032475162377549690981884983618595036

5/29:

0.3461392977978521670665682986277862391250376411142008809038801004379433399571578232156439132575845381

6/29:

0.3968318587024627677784980108360430782451064534010180924192774621655367616066891061214489144941909209

7/29:

0.4452968953156480996199589860200447208894592920343137828361361727506558285036533487825544822712796275

8/29:

0.4866668720248198075630960041003552540579119386089242928762167992026351307684846839805845743819451634

9/29:

0.5315080003164396118450780002391990138632720642348034758910207188013436400052004178790865564214030390

10/29:

0.5765518820451163518590257141301549028368103173670470070580331481615803053446747469408335879902342834

11/29:

0.6073626403856100572267670325862387204556847332740823818628959176015909112192863910960174318805828066

12/29:

0.6445725157563080078105725941553078984903775515591101929612330286800904557869069123885521592958955583

13/29:

0.6847489730860354835001573751290130807400209053142771162568233586268393725229786503034245138576504630

14/29:

0.7342675045476539300639372043817118983001292967539901530971450350637131384055466787705738042489887430

15/29:

0.7676728557855525841535211617554929877222927582384595018676201650526651549069716021059426941398334151

16/29:

0.7836000892498630231322879065834132279436188769714365950083691048204509580635538244028642775681184149

17/29:

0.8077978940634136453849022547009357250370733354464325923214195563319947472089021701384592540973627543

18/29:

0.8349865276859460132350195075336378691109999865091865042080309670228366247328159654059068293413243464

19/29:

0.8667698632103241986784261003380835987280596175166738991780407113521679432288954277934756519485659429

20/29:

0.8932088753010485852033732951681978099892928983653822195478149174798657460582844530407140008746478972

21/29:

0.9205333438149398589114550144513659983219305506031537452624492379948346986484082151084794479755030451

22/29:

0.9513564186137004210180957477265075433981472140291894404247546040198598656094341151156632481665270415

23/29:

0.9817191417638892838269733385136592331072410310515955299319122161311553480859725898618011325912890547

24/29:

1.014420225752314433293051129003097388065652980104429252033563266076806098795207055069703486128583761

25/29:

1.049792167965731262827086670926913635168038753956227309763927972651482453135006905914859387998887707

26/29:

1.088681970294322116960111609328725596299504530365364686061847035578400721036751941570704880332888396

27/29:

1.132713897107366543987175461402828929640131614304576584687448500030298370844487435950972543425439500

28/29:

1.185064736742255453982837117423704829501096048772433105567726769266843038192405858872503409488733092

29:

2.818705068162265891837705443768942456068191301594464640392711779027124876428301854397632231177916308

29/2:

2.466225556070975952436622449823872424441553127491430503463182649321712742931174089041953386578866008

29/3:

2.261934718384462026153710529686857343412204895519680206780449660920677939323810903975954927856240477

29/4:

2.118756651040388664073391829105681927719551525028187808691590249723544492723723525961866613123481401

29/5:

2.007607927227542568986096132041160186925218318462365109242504582540071371751515374650734696893990321

29/6:

1.918020650395236710929407052374208211518014524771587446693174400466761014432330679587003086721922785

29/7:

1.844801423450542126996972942083042415113474209856442814606849858538431289515135302099154283695301314

29/8:

1.764167411089971802416223014863787795959930777457350561676285897109552349035756979429619082134551217

29/9:

1.712636889908527638167473556326307933559432206978811200093288982804329506683423568721501126246743126

29/10:

1.672000457930837420391174570977449218226749920364436320468296129668582885499556766128417405171679422

29/11:

1.601228779198426514506931267727356626655896114995308097638543782767830584123573212889500502230627399

29/12:

1.557716913077744352208883769208660754685079082934516299656313152643551934818358371605667718298414266

29/13:

1.527516939961156078577274144518567641650815865701079720880605953859872446397413912215331607836297187

29/14:

1.520982687991568855132441351933546075050267828990408174272657572631977215268632406024760023087190968

29/15:

1.484167521185401662696807579393953109596432665927688370277398985768485966153478430738155875337011269

29/16:

1.420275161765376729427271830682436475647809214510728828452669002487067361490191306730191503092214627

29/17:

1.378008172225823277421303846254537413298536866349796775136539243154579274650480172589136374636677640

29/18:

1.345256072382913021323086984359749900234388867153689367890716557981236784291759055376183225049911447

29/19:

1.322964528057863250614439837358127598058617310946502267166483191011203702823050916105831258237284860

29/20:

1.295152869186520448544891277993886824484474702629804218344331630345805331784512456909035301268239451

29/21:

1.271212712887297900401533115194743521492189807975783743457667995326200298133516106578376380537599443

29/22:

1.254060733627150554978398940184941761752103145765749716923540159844360731939708606288828827128603827

29/23:

1.237819787441425618738357687691135554787390865238968276870671924687108917151878482869227515006407939

29/24:

1.225757772784046606895770114212076010579330684292852012873888946509474036044208524875891712405372044

29/25:

1.217758914840248264879420538275219816794924954589223679326156448275719645636608010861236890078709740

29/26:

1.214299120712897745840124487328193934334062745407522149838214001222062342694838704059632366525144749

29/27:

1.216618630226430732430669940025260702206067289438248924293926166699209361277412431206600139234731315

29/28:

1.227388477340193148767938443045980001983278050514305716480859868169230289556420353832235674113330703

1/30:

0.09451661063324492469256565998690750819453906386908844935440775649743025622830428836751050453465783887

7/30:

0.4323191129690012257827876964442758451165394917905401794775774653929988829160731790288462741336987698

11/30:

0.5945614958147622123120373683471705038723943343699818606830811585245204342663770791718056017691560478

13/30:

0.6667814330078367130968293203786356317505801400448811499373796018554849186431845568302310606450798838

17/30:

0.7935817861746830925654199862719100895294401554457374275451283741888449079267268829028062110397456800

19/30:

0.8456704951636740116600371369430865526082413485753893747290170007255637248623741281007997091963841149

23/30:

0.9577284085071399197655992878691573046589558095987354860987752880801399998957360326258191311762107414

29/30:

1.187029253475694653421468374039035742219754809356602023082014701542281497498143135435271062695066428

30:

2.835498318997347740776969799607225245836171916072653480632232694922907686849128651025315136039735166

30/7:

1.852796198438576681926232984761182193356597821959457912046760565969995212497456481552198317715851871

30/11:

1.621531352222078760851011004583192283288348184645405074590221341430510275271937488650378913915880130

30/13:

1.538726383864238568684990739335312996347492630872802653701645235051119043022733592685148601488645886

30/17:

1.400438446190617222174270564009253099169600274316007225079638307392079249282459205122599195952492376

30/19:

1.335269202890011597357953374120662977802486339855877960098447895882469039256380202264420593467974918

30/23:

1.249210967618008590998607766785857353902985838607046286215793854017573912907481781685851040664622706

30/29:

1.227961296698994469056691421419692147123884285541312437671049691250636031894630829760625237270758374

Definition

The multiplicative autocorrelation of the fractional-part function is $A(\lambda)=\int_0^\infty \{t\}\{\lambda t\}\,dt/t^2$ for $\lambda>0$, where $\{x\}=x-\lfloor x\rfloor$ [1]. This table records $A(\lambda)$ at positive rational dilations.

Parameters

$\lambda$

— Dilation ($\lambda\in\mathbb Q$, $\lambda>0$)

Formulas

(1)

$A(\lambda)=\lambda A(1/\lambda)$ for $\lambda>0$ [1], Proposition 85.

(2)

$A(1)=C=\log(2\pi)-\gamma$ [1], Proposition 87.

(3)

$A(p/q)=\frac{1-p/q}{2}\log(p/q)+\frac{1+p/q}{2}C-\frac{\pi}{2q}\bigl(V(p,q)+V(q,p)\bigr)$, where $V(p,q)=\sum_{m=1}^{q-1}\{mp/q\}\cot(\pi m/q)$ and $V(p,1)=0$ [1], Proposition 89. For $q>1$, $V(p,q)=-c_0(\bar p/q)$, where $p\bar p\equiv1\pmod q$ and $1\leq\bar p<q$, and $c_0$ is the Bettin–Conrey cotangent sum [2], §2.

(4)

$A(p/q)=\frac1q\int_0^1\{qu\}\{pu\}\,\psi^{(1)}(u)\,du$, where $\psi^{(1)}(u)=\sum_{n=0}^\infty(n+u)^{-2}$ is the trigamma function. The formula follows by splitting the defining integral into periods of length $q$; the integrand has a finite limit at $u=0$.

(5)

$\langle\rho_j,\rho_k\rangle=\frac1j A(j/k)$ for positive integers $j,k$. Indeed, $t=1/(jx)$ gives $\int_0^\infty\{1/(jx)\}\{1/(kx)\}\,dx=\frac1j\int_0^\infty\{t\}\{(j/k)t\}\,dt/t^2$.

Comments

(6)

All logarithms are natural, and $\pi$ is positive. Write $\lambda=p/q$ with coprime positive integers $p,q$, and put $C=\log(2\pi)-\gamma$, where $\gamma$ is the Euler constant. Ordinary fractional parts are used, including value zero at integers; endpoint values do not affect the integral.

(7)

For $\rho_n(x)=\{1/(nx)\}$ in $L^2((0,\infty),dx)$, the values of $A$ supply the Gram matrix of the integer dilates in the strengthened Nyman–Beurling criterion [3]. The inner product uses the full positive half-line, with no anchoring, coefficient constraint, or normalization of the basis vectors. These finite integral evaluations require no assumption about the Riemann hypothesis.

(8)

The Bettin–Conrey cotangent sums give $A(p/q)$ through (3). The Vasyunin sum is also spelled Vassiounine sum; its conversion uses a modular inverse, rather than the same numerator [2].

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

R = RealBallField(numberdb.bits(100, losing=128))
def V(p, q):
    return sum((R(QQ((m*p) % q)/QQ(q)) *
                (R.pi()*R(QQ(m)/QQ(q))).cot()
                for m in range(1, q)), R(0))
def A(p, q):
    lam = QQ(p)/QQ(q)
    p, q = int(lam.numerator()), int(lam.denominator())
    x = R(lam)
    C = (2*R.pi()).log() - R.euler_constant()
    return ((1-x)*x.log() + (1+x)*C)/2 - R.pi()*(V(p,q)+V(q,p))/(2*q)

print(A(2, 3))

References

[1]

L. Báez-Duarte, M. Balazard, B. Landreau and E. Saias, Sur l’autocorrélation multiplicative de la fonction « partie fractionnaire », 2003, §§4 and 9, especially Propositions 85, 87 and 89. (arXiv)

[2]

S. Bettin and B. Conrey, Period functions and cotangent sums, Algebra & Number Theory 7 (2013), no. 1, 215–242, §2. (arXiv) (doi)

[3]

L. Báez-Duarte, A strengthening of the Nyman–Beurling criterion for the Riemann Hypothesis, 2002, §1 and Theorem 1.1. (arXiv)

Similar tables

Bettin–Conrey cotangent sums — The modular-inverse sums $c_0(\bar p/q)$ provide the arithmetic terms in (3).

Values of the polygamma functions at rational numbers — The trigamma function $\psi^{(1)}(u)$ is the kernel in (4).

Stieltjes constants — The entry $\gamma_0=\gamma$ enters $A(1)=\log(2\pi)-\gamma$.

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every reduced positive rational $p/q$ with $1\leq p,q\leq30$, exactly the distinct ratios needed for the Gram matrices of the first 30 integer dilates in (7). Both reciprocal dilations are retained because their integral values have different scales. Thirty is a practical, noncanonical initial bound for the shared finite approximation problem, not an intrinsic threshold. The order is increasing $\max(p,q)$, then $p$, then $q$; the argument-selection attachment explains the coverage)

How they were obtained:

Computed from the finite Vasyunin-sum formula (3) using Arb real balls at 461 working bits, with exact rational arguments. Every value is finite, retains at least 450 bits of relative accuracy, and has absolute radius less than $4.44\times10^{-136}$; these enclosures justify the 100 significant digits stored. No rounded values are imported from companion tables.

Every selected value also agrees with an independent 512-bit evaluation of (4). That checker integrates the piecewise quadratic fractional-part product analytically against 16 terms of the trigamma series and 101 terms of a uniformly convergent expansion for the remainder, then adds an explicit bound for the omitted tail. Its largest enclosure radius is less than $1.12\times10^{-122}$. Both enclosures lie strictly within each stored last-place interval. Known-answer and deliberately perturbed controls precede the comparisons. Reciprocal symmetry, the unit value, all 277 modular-inverse cotangent conversions and all 900 Gram symmetries through index 30 are also checked. The generator, checks, derivation and numerical evidence are attached.