Exploring the Riemann Zeta function : 190 years from Riemann's birth

300 Pages·2017·5.007 MB·English

by  Montgomery, Hugh, Nikeghbali, Ashkan, Rassias, Michael Th

Hugh Montgomery · Ashkan Nikeghbali
Michael Th. Rassias Editors
Exploring the
Riemann Zeta
Function
190 years from Riemann’s Birth
With a preface by
Freeman J. Dyson
Exploring the Riemann Zeta Function
Hugh Montgomery • Ashkan Nikeghbali
Michael Th. Rassias
Editors
Exploring the Riemann Zeta
Function
190 years from Riemann’s Birth
With a preface by
Freeman J. Dyson
123
Editors
HughMontgomery AshkanNikeghbali
DepartmentofMathematics InstitutfürMathematik
UniversityofMichigan UniversitätZürich
AnnArbor,MI,USA Zürich,Switzerland
MichaelTh.Rassias
InstitutfürMathematik
UniversitätZürich
Zürich,Switzerland
ISBN978-3-319-59968-7 ISBN978-3-319-59969-4 (eBook)
DOI10.1007/978-3-319-59969-4
LibraryofCongressControlNumber:2017947901
MathematicsSubjectClassification(2010):11-XX,30-XX,33-XX,41-XX,43-XX,60-XX
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Preface by Freeman J. Dyson:
Quasi-Crystals and the Riemann Hypothesis
Ontheoccasionofthe190yearsfromRiemann’sbirth,whichledtotheconception
of this book, I am presenting a challenge to young mathematicians to study
quasicrystals and use them to prove the Riemann Hypothesis. Among the readers,
thosewhoareexpertsinnumbertheorymayconsiderthechallengefrivolous.Those
whoarenotexpertsmayconsiderituninterestingorunintelligible.Nevertheless,I
amputtingitforwardforyourseriousconsideration.Oneshouldalwaysbeguided
bythewisdomofthephysicistLeoSzilard,whowrotehisownTenCommandments
to stand beside those of Moses. Szilard’s second commandment says, “Let your
actsbedirectedtowardaworthygoal,butdonotaskiftheywillreachit;theyare
to be models and examples, not means to an end.” The Riemann Hypothesis is a
worthy goal, and it is not for us to ask whether we can reach it. I will give you
somehintsdescribinghowitmightbeachieved.Here,Iwillbegivingvoicetothe
mathematicianthatIwasalmost70yearsagobeforeIbecameaphysicist.
There were until recently two supreme unsolved problems in the world of pure
mathematics, the proof of Fermat’s Last Theorem and the proof of the Riemann
Hypothesis. In 1998, my former Princeton colleague Andrew Wiles polished off
Fermat’s Last Theorem, and only the Riemann Hypothesis remains. Wiles’ proof
oftheFermatTheoremwasnotjustatechnicalstunt.Itrequiredthediscoveryand
explorationofanewfieldofmathematicalideas,farwiderandmoreconsequential
thantheFermatTheoremitself.ItislikelythatanyproofoftheRiemannHypothesis
willlikewiseleadtoadeeperunderstandingofmanydiverseareasofmathematics
andperhapsofphysicstoo.TheRiemannHypothesissaysthatonespecificfunction,
thezeta-functionthatRiemannnamed,hasallitscomplexzerosuponacertainline.
Riemann’s zeta-function and other zeta-functions similar to it appear ubiquitously
in number theory, in the theory of dynamical systems, in geometry, in function
theory, and in physics. The zeta-function stands at a junction where paths lead in
manydirections.Aproofofthehypothesiswillilluminatealltheconnections.Like
every serious student of pure mathematics, when I was young, I had dreams of
provingtheRiemannHypothesis.IhadsomevagueideasthatIthoughtmightlead
to a proof, but never pursued them vigorously. In recent years, after the discovery
v
vi PrefacebyFreemanJ.Dyson:Quasi-CrystalsandtheRiemannHypothesis
of quasicrystals, my ideas became a little less vague. I offer them here for the
consideration of any young mathematician who has ambitions to win a Fields
Medal.
Now,IjumpfromtheRiemannHypothesistoquasicrystals.Quasicrystalswere
one of the great unexpected discoveries of recent years. They were discovered in
twoways,asrealphysicalobjectsformedwhenalloysofaluminumandmanganese
aresolidifiedrapidlyfromthemoltenstateandasabstractmathematicalstructures
inEuclideangeometry.Theywereunexpected,bothinphysicsandinmathematics.
Iamherediscussingonlytheirmathematicalproperties.
Quasicrystalscanexistinspacesofone,two,orthreedimensions.Fromthepoint
of view of physics, the three-dimensional quasicrystals are the most interesting,
since they inhabit our three-dimensional world and can be studied experimentally.
Fromthepointofviewofamathematician,one-dimensionalquasicrystalsaremore
interesting than two-dimensional quasicrystals because they exist in far greater
variety, and the two-dimensional are more interesting than the three-dimensional
forthesamereason.Themathematicaldefinitionofaquasicrystalisasfollows.A
quasicrystalisadistributionofdiscretepointmasseswhoseFouriertransformisa
distributionofdiscretepointfrequencies.Ortosayitmorebriefly,aquasicrystalis
apurepointdistributionthathasapurepointspectrum.Thisdefinitionincludesasa
specialcasetheordinarycrystalswhichareexactlyperiodicpointdistributionswith
exactlyperiodicpointspectra.
Excludingtheordinarycrystals,quasicrystalsinthreedimensionscomeinvery
limited variety, all of them being associated with the icosahedral rotation group.
The two-dimensional quasicrystals are more numerous, roughly one distinct type
associatedwitheachregularpolygoninaplane.Thetwo-dimensionalquasicrystal
with pentagonal symmetry is the famous Penrose tiling of the plane, discovered
by Penrose before the general concept of quasicrystal was invented. Finally, the
one-dimensional quasicrystals have a far richer structure since they are not tied
to any rotational symmetries. So far as I know, no complete enumeration of one-
dimensional quasicrystals exists. It is known that a unique quasi-crystal exists
correspondingtoeveryPisot-VijayaraghavannumberorPVnumber.APVnumber
isarealalgebraicinteger,arootofapolynomialequationwithintegercoefficients,
such that all the other roots have absolute value less than one (see [1]).The set of
all PV numbers is infinite and has a remarkable topological structure. The set of
allone-dimensionalquasicrystalshasastructureatleastasrichasthesetofallPV
numbersandprobablymuchricher.Wedonotknowforsure,butitislikelythata
hugeuniverseofone-dimensionalquasicrystalsnotassociatedwithPVnumbersis
waitingtobediscovered.
Here comes the punch line of my argument, the essential point linking one-
dimensionalquasicrystalswiththeRiemannHypothesis.IftheRiemannHypothesis
is true, then the zeros of the zeta-function form a one-dimensional quasicrystal
according to the definition. They constitute a distribution of point masses on a
straightline,andtheirFouriertransformislikewiseadistributionofpointmasses,
oneateachofthelogarithmsofordinaryprime-powernumbers.AndrewOdlyzko
(see [2]) has published a beautiful computer calculation of the Fourier transform
PrefacebyFreemanJ.Dyson:Quasi-CrystalsandtheRiemannHypothesis vii
of the zeta-function zeros. The calculation shows precisely the expected structure
oftheFouriertransform,withasharpdiscontinuityateverylogarithmofaprime-
powernumberandnowhereelse.
My proposal is the following. Let us pretend that we do not know that the
Riemann Hypothesis is true. Let us tackle the problem from the other end. Let
us try to obtain a complete enumeration and classification of all one-dimensional
quasicrystals. That is to say, we enumerate and classify all point distributions that
have a discrete point spectrum. We shall then find the well-known quasicrystals
associated with PV numbers and also a whole slew of other quasicrystals, known
andunknown.Amongtheslewofotherquasicrystals,weshouldbeabletoidentify
onecorrespondingtotheRiemannzeta-functionandonecorrespondingtoeachof
theotherzeta-functionsthatresembletheRiemannzeta-function.Supposethatwe
canproverigorouslythatoneofthequasicrystalsinourenumerationhasproperties
that identify it with the zeros of the Riemann zeta-function. Then we have proved
theRiemannHypothesisandcanwaitforthetelephonecallannouncingtheaward
oftheFieldsMedal.
These are of course idle dreams. The problem of classifying one-dimensional
quasicrystalsishorrendouslydifficult,probablyatleastasdifficultastheproblems
that Andrew Wiles took 7 years to fight his way through. But still, the history
of mathematics is a history of horrendously difficult problems being solved by
young people too ignorant to know that they were impossible. The classification
ofquasicrystalsisaworthygoalandmighteventurnouttobeachievable.Problems
of that degree of difficulty will not be solved by old men like me. Let the young
peoplenowhaveatry.
InstituteforAdvancedStudy FreemanJ.Dyson
1EinsteinDr
Princeton,NJ08540,USA
References
1.M.J.Bertinetal.,PisotandSalemNumbers(BirkhäuserVerlag,Basel,1992)
2.A.M. Odlyzko, Primes, quantum chaos and computers, in Number Theory, Proceedings of a
Symposium(NationalResearchCouncil,Washington,DC,1990),pp.35–46
Contents
PrefacebyFreemanJ.Dyson:
Quasi-CrystalsandtheRiemannHypothesis................................. v
An Introduction to Riemann’s Life, His Mathematics,
andHisWorkontheZetaFunction............................................ 1
RogerBaker
Ramanujan’sFormulafor(cid:2)(2nC1)........................................... 13
BruceC.BerndtandArminStraub
Towards a Fractal Cohomology: Spectra of Polya–Hilbert
Operators,RegularizedDeterminantsandRiemannZeros................. 35
TimCoblerandMichelL.Lapidus
TheTemptationoftheExceptionalCharacters............................... 67
JohnB.FriedlanderandHenrykIwaniec
Arthur’sTruncatedEisensteinSeriesforSL(2,Z)andtheRiemann
ZetaFunction:ASurvey ........................................................ 83
DorianGoldfeld
OnaCubicMomentofHardy’sFunctionwithaShift ...................... 99
AleksandarIvic´
SomeAnaloguesofPairCorrelationofZetaZeros .......................... 113
YunusKarabulutandCemYalçınYıldırım
Bagchi’sTheoremforFamiliesofAutomorphicForms...................... 181
E.Kowalski
TheLiouvilleFunctionandtheRiemannHypothesis........................ 201
MichaelJ.MossinghoffandTimothyS.Trudgian
ExplorationsintheTheoryofPartitionZetaFunctions..................... 223
KenOno,LarryRolen,andRobertSchneider
ix
x Contents
ReadingRiemann ................................................................ 265
S.J.Patterson
ATaniyamaProductfortheRiemannZetaFunction ....................... 287
DavidE.Rohrlich
An Introduction to Riemann’s Life, His
Mathematics, and His Work on the Zeta
Function
RogerBaker
Abstract Although the zeta function was first defined and used by Euler, it is to
Bernhard Riemann, in an article written in 1859, that we owe our view of the
zeta function as a meromorphic function in the plane with a functional equation.
Riemann is a very remarkable figure in the history of mathematics. The present
articledescribeshiscareerincludingthemajormathematicalhighlights,andgives
somediscussionofhispublishedandunpublishedworkonthezetafunction.
1 Introduction
In 1846 a frail and painfully shy youth of nineteen arrived in Göttingen, in the
Kingdom of Hanover, to begin his university studies. Bernhard Riemann was to
spend most of his adult life in Göttingen as an undergraduate, doctoral candidate,
Privatdozent (junior university instructor), extraordinary professor (roughly,
associate professor), and full professor. The best of his research papers, written
inthe1850s,wereamongtheverymostinfluentialpapersinthemathematicsofhis
century.Inparticular,inapaperwrittenin1859,hesetEuler’szetafunction(ashe
would have referred to it) on a modern footing. Today, zeta functions are perhaps
the part of number theory that we would most like to understand better, as indeed
thepublicationofthepresentvolumetestifies.
Georg Friedrich Bernhard Riemann was born on 17 September 1826 in
Breselenz, in the Kingdom of Hanover. His father Friedrich was the local pastor;
later he took over the parish of Quickborn, not far from the Free Imperial City of
Hamburg,butstillintheKingdomofHanover.Bernhardwastutoredbyhisfather
uptotheageofthirteen,whenhemoved tothecityofHanover; helivedwithhis
grandmother and attended high school for 2 years. After her death, he attended
thehighschoolinLüneburg,whichissouth-eastofHamburg.Hewasnotnotably
successful except in mathematics. (At this time he read Legendre’s Théorie des
Nombres[12].)
R.Baker((cid:2))
DepartmentofMathematics,BrighamYoungUniversity,Provo,UT84602,USA
e-mail:[email protected]
©SpringerInternationalPublishingAG2017 1
H.Montgomeryetal.(eds.),ExploringtheRiemannZetaFunction,
DOI10.1007/978-3-319-59969-4_1