Table Of ContentOxfordGraduateTextsinMathematics
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
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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.
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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)
