Experimental number theory PDF
Preview Experimental number theory
OxfordGraduateTextsinMathematics SeriesEditors R.Cohen S.K.Donaldson S.Hildebrandt T.J.Lyons M.J.Taylor OXFORDGRADUATETEXTSINMATHEMATICS Booksintheseries 1. KeithHannabuss:Anintroductiontoquantumtheory 2. ReinholdMeiseandDietmanVogt:Introductiontofunctional analysis 3. JamesG.Oxley:Matroidtheory 4. N.J.Hitchin,G.B.Segal,andR.S.Ward:Integrablesystems: twistors,loopgroups,andRiemansurfaces 5. WulfRossmann:Liegroups:Anintroductionthroughlineargroups 6. QingLiu:Algebraicgeometryandarithmeticcurves 7. MartinR.BridsonandSimonM.Salamon(eds):Invitationsto geometryandtopology 8. ShumelKantotovitz:Introductiontomodernanalysis 9. TerryLawson:Topology:Ageometricapproach 10. MeinolfGeck:Anintroductiontoalgebraicgeometryandalgebraic groups 11. AlastairFletcherandVladimirMarkovic:Quasiconformalmapsand Teichmüllertheory 12. DominicJoyce:Riemannianholonomygroupsandcalibrated geometry 13. FernandoRodriguezVillegas:ExperimentalNumberTheory Experimental Number Theory Fernando Rodriguez Villegas 1 3 GreatClarendonStreet,Oxfordox26pd OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwidein Oxford NewYork Auckland CapeTown DaresSalaam HongKong Karachi KualaLumpur Madrid Melbourne MexicoCity Nairobi NewDelhi Shanghai Taipei Toronto Withofficesin Argentina Austria Brazil Chile CzechRepublic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore SouthKorea Switzerland Thailand Turkey Ukraine Vietnam OxfordisaregisteredtrademarkofOxfordUniversityPress intheUKandincertainothercountries PublishedintheUnitedStates byOxfordUniversityPressInc.,NewYork ©OxfordUniversityPress,2007 Themoralrightsoftheauthorhavebeenasserted DatabaserightOxfordUniversityPress(maker) Firstpublished2007 Allrightsreserved.Nopartofthispublicationmaybereproduced, storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans, withoutthepriorpermissioninwritingofOxfordUniversityPress, orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriate reprographicsrightsorganization.Enquiriesconcerningreproduction outsidethescopeoftheaboveshouldbesenttotheRightsDepartment, OxfordUniversityPress,attheaddressabove Youmustnotcirculatethisbookinanyotherbindingorcover andyoumustimposethesameconditiononanyacquirer BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressCataloginginPublicationData Dataavailable TypesetbyNewgenImagingSystems(P)Ltd.,Chennai,India PrintedinGreatBritain onacid-freepaperby BiddlesLtd.,King’sLynn,Norfolk ISBN978–0–19–852822–7 ISBN978–0–19–922730–3(pbk) 1 3 5 7 9 10 8 6 4 2 To Lula, Malena and Adriana This page intentionally left blank Acknowledgements Numerous people have contributed to this book, in one way or another. I thankthemall. Firstandforemost,IwouldliketothankKarimBelabas,themainmain- tainer of PARI-GP, who read the book line by line and suggested many significant improvements both in the text and in the coding of the scripts. Hishelpwasinvaluable(remainingbugsareofcoursemine).Iwouldalso like to thank: Don Zagier, from whom I learned, over the years, a lot of mathematics both theoretical and experimental; John Tate, for our many conversations, mathematical and otherwise, and for his many suggestions andcommentsonthemanuscript;DavidBoyd,experimentalnumbertheo- ristparexcellence,whoseworkIusedinmanyexamplesinthebook;Paul Gunnells,foracarefulreadingofanearlierversion;andMargaretCombs, forthefiguresinthebookandhergeneralTeXpertadvice. Specialthanksgotomyfamilyfortheirlove. This page intentionally left blank Preface InMay30,1799Gausswroteinhismathematicaldiary √ Terminum medium arithmetico–geometricum inter 1 et 2 esse = π usque ad (cid:2) figuramundecimamcomprobavimus,quaredemonstrataprorsusnovuscampusin analysiscertoaperietur. whichtranslatesto(see[27]p.281) √ π We have established that the arithmetic–geometric mean between 1 and 2 is (cid:2) to the eleventh decimal place; the demonstration of this fact will surely open an entirelynewfieldofanalysis. Thearithmetic–geometricmeanM(a,b)oftwopositiverealnumbersa,b isthecommonlimitofthesequencea ,b definedbytherecursion n n (cid:1) an+1 := 12(an+bn), bn+1 := anbn, a0 :=a, b0 :=b. (0.1) Thesymbol(cid:2) wasGauss’sspecialnotationfortheintegral (cid:2) 1 1 (cid:2) :=2 (cid:1) dx, (0.2) 0 1−x4 whichhadappearedina1691paperofJ.Bernoulli;thenumericalvalue (cid:2) π 1 x2 =2 (cid:1) dx=1.19814023473559223... (0.3) (cid:2) 0 1−x4 was well known to Gauss after its calculation to 17 decimals by J. Stirling in1730. ThereisnodoubtthatGauss’sremarkwasprophetic(see[27]forabeau- tifulmodernrenditionofthisentirelynewfieldthathewastodiscover).In fact,numericalexperimentationiscrucialtonumbertheory,perhapsmore sothantootherareasofmathematics.WitnesstheBirch–Swinnerton-Dyer conjecture, one of the outstanding million-dollar millennium problems of the Clay Mathematics Institute, which was inferred from numerical calcu- lations. Indeed, as Cassels has said, to a large degree number theory is an experimentalscience. Here is what Hermite wrote to Stieltjes in his letter of March 12, 1884 (inmytranslationfromthefrenchoriginal,see[4]letter47)
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