From Arithmetic to Zeta-Functions: Number Theory in Memory of Wolfgang Schwarz

552 Pages·2016·8.59 MB·English

by  Jürgen Sander

Jürgen Sander
Jörn Steuding
Rasa Steuding Editors
From Arithmetic
to Zeta-Functions
Number Theory in Memory
of Wolfgang Schwarz
From Arithmetic to Zeta-Functions
JuRrgen Sander • JoRrn Steuding (cid:129) Rasa Steuding
Editors
From Arithmetic
to Zeta-Functions
Number Theory in Memory
of Wolfgang Schwarz
123
Editors
JuRrgenSander JoRrnSteuding
InstitutfürMathematikundAngewandte InstitutfürMathematik
Informatik UniversitätWürzburg
UniversitätHildesheim WuRrzburg,Germany
Hildesheim,Germany
RasaSteuding
InstitutfürMathematik
UniversitätWürzburg
WuRrzburg,Germany
ISBN978-3-319-28202-2 ISBN978-3-319-28203-9 (eBook)
DOI10.1007/978-3-319-28203-9
LibraryofCongressControlNumber:2016947422
Mathematics Subject Classification (2010): 11Axx, 11Bxx, 11Dxx, 11Jxx, 11Kxx, 11Lxx, 11Mxx,
11Nxx,11Pxx,01Axx
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Publishedwithkindpermissionof©JürgenWolfart,2009
DasschönsteDenkmal,dasein Mensch
bekommen kann,stehtin denHerzen seiner
Mitmenschen.1
AlbertSchweitzer
1Themostbeautifulmemorialwhichcanbeerectedforsomebodyistheimprintheleavesbehind
intheheartsofothers.
Wenn er aber studiert, müssen ihm bedeutende
ideale Ziele vorschweben
The headline above is a passage from a personal statement written by Wolfgang
Schwarzshortlyafterfinishingschool,explaininghisviewsandgoalswhilelooking
forwardto takingup studies at the Universityof Erlangen.Literally, it says that a
student’smotivationmustbesubstantialandidealistic.Atthattimeitwascustomary
towriteaproposalforenrollmentataGermanuniversity.
Fig.1 Left:the17-year-oldWolfgangSchwarz;right:excerptfromhisessay.Publishedwithkind
permissionof©DorisSchwarz,2013
AsaprofessionIaimatthepositionofahighschoolteacherofmathematicsandphysics,
or,ifpossible,thecareerofauniversityprofessorofmathematics.2
This is another quotation from Wolfgang Schwarz’s essay showing a rather self-
confidentyoungmanwithambitiousgoals.Andindeed,thispromisingyoungman
liveduptohisownexpectationsandbecameanoutstandingmathematician.Hisfield
ofresearchwasnumbertheorywithanemphasisonanalysis.Yet,hisinterestswere
as widespread as his knowledge, ranging from elementary to algebraic aspects of
numbertheory,andfromanalysistoalgebra.Thisbroadspectrumiswellreflected
2ThisisourtranslationoftheGermanoriginal(seeFig.1).PleasenotethattheGermanhöheres
Lehramt standsforteachingpupilsfrom grade 6to13atagymnasium, andHochschule atthat
timewasequivalenttouniversity.
vii
viii Wenneraberstudiert,müssenihmbedeutendeidealeZielevorschweben
in the variety of research articles and surveys in this memorial volume, some of
themenrichedwithpersonalreminiscences.
WolfgangSchwarzwasbornonthe21stofApril1934inSelb(notfarfromHof
innorthernBavaria),whichiswellknownforitsfamousporcelainmanufactories.
His father was a teacher at a primary school and, although he died untimely
in 1945 during World War II, he left a deep impression on young Wolfgang,
which is best manifested in his early curiosity about mathematics and natural
sciences. Their mother raised Wolfgang and his brother Werner and encouraged
theirmusicaltalents. In 1951Schwarz obtainedhis universityentrancecertificate,
calledReifezeugnisinGerman.HeenrolledattheFriedrich-Alexander-Universität
Erlangento study mathematicsandphysics;amonghisteacherswere Otto Haupt,
Georg Nöbeling, Wilhelm Specht, and, most importantly, the eminent Theodor
Schneider,whoisfamousforhiselegantsolutionoftheseventhHilbertproblemon
thetranscendenceofnumbersoftheform˛ˇ,where˛isalgebraicandˇirrational.3
After passing the state examination for teachers in mathematics and physics in
1956 Wolfgang Schwarz continued his studies as an assistant of Schneider, first
in Erlangen and later at the Albert-Ludwigs-Universität Freiburg (Fig.2). His
Fig.2 WolfgangSchwarzduringhisdoctoratestudies.Publishedwithkindpermissionof©Doris
Schwarz,2013
doctorate thesis of 1959 dealt with questions about representations of integers as
sumsofpowersofprimenumbers.Forhishabilitation,4accomplishedin1964also
atFreiburg,hestudiedso-calledTauberiantheoremswithapplicationstopartitions.
Thetopicsandmethodsusedinbothofhisthesesformacentralpartoftwentieth-century
analyticnumbertheory.Inthismemorialvolume,honoringthelifeandworkofWolfgang
3Independently,AlexanderGelfondprovidedanothersolutionaboutthesametime.
4Habilitationisadegreefollowingthedoctorate,atthattimenecessarytoobtainthepermission
forindependentteaching(venialegendi)andforappointmentasaprofessor.
Wenneraberstudiert,müssenihmbedeutendeidealeZielevorschweben ix
Schwarz, the reader will find several contributions closely related to Schwarz’s work,
e.g. those by Jörg Brüdern, Rainer Dietmann & Christian Elsholtz, and Rebecca Ulrike
Jakob, allrelyingonthecirclemethod,aswellasAleksandarIvic´’sarticleonTauberian
theorems.SomeofthesenewresultsuseorextendpreviousonesobtainedbySchwarz.In
thebackground ofJánosPintz’sarticleonrecentprogress inthetheoryofprimenumber
distributionaresievemethods,whichisanotherdirectionofresearchwhereSchwarzmade
significantcontributionsandpublishedmonographsatanearlyage.
In the late 1960s Wolfgang Schwarz was appointed professor at the Albert-
Ludwigs-Universität Freiburg, and in 1969 he became Ordinarius at the Johann-
Wolfgang-Goethe-UniversitätFrankfurt. In 1974, he and his family settled in the
village of Ruppertshain near Königstein in the lovely Taunus mountain range
north of Frankfurt. The Schwarz family quickly became acquainted with its new
surroundingsandmademanyfriends.
Wolfgang Schwarz’s mathematical ideas and inventions were creative and
innovative.Hisbroadknowledgetogetherwithhismasteryofadvancedtechniques
wasandstillisveryimpressive.Healwayshadadesireforsimplicityandelegance.
Thisiswellreflectedinhisnumerous(morethanonehundred)researcharticles(see
thecompletelistofhispublicationssubsequenttothispreface).Afewofthemdeal
withelementary,diophantineandtranscendentalquestions.
The articles of Régis de la Bretèche & Gérard Tenenbaum, Štefan Porubský, Andrzej
Schinzel, Jan-Christoph Schlage-Puchta, Tarlok Shorey & Rob Tijdeman, and Jürgen
Spilkerareelementaryinnatureandlieatthecoreofnumbertheory.Diophantineaspectsor
transcendencetopicsarethemajorthemeofthecontributionsbyIouliaBaoulina&Pieter
Moree, Valentin Blomer, Yann Bugeaud, Peter Bundschuh & Keijo Väänänen, Carsten
Elsner,ManfredMadritsch&RobertTichy,andJörnSteuding;someofthemaredirectly
relatedtoresultsofSchwarz.
OneofWolfgangSchwarz’smajorresearchinterestswasthestudyofarithmeti-
calfunctions.TogetherwithhiscolleagueandfriendJürgenSpilkerfromFreiburg,
Schwarzpublishedthestandardreferenceinthisfield,continuingimportantworks
by Wintner, Delange, Wirsing, Elliott, and Halász. Methods of probabilistic and
functional-analytic flavor form an essential part of Schwarz & Spilker’s Arith-
metical Functions and provide a rather new perspective on this classical topic.
In particular, the concept of Ramanujan expansions, studies of so-called related
functions, and Gelfand’s theory of maximal ideal spaces make reading this book
agreatpleasure.
Inthismemorialvolumeseveralexamplesofrecentandpastworkonarithmeticalfunctions
canbefoundinthearticlesofPeterElliott,Karl-HeinzIndlekofer, AleksandarIvic´,Lutz
Lucht, Eugenijus Manstavicˇius, and Friedemann Tuttas, some of them directly related
to Wolfgang Schwarz’s work: the contribution by Aleksandar Ivic´ includes aspects of
Schwarz’s study of the value-distribution of certain arithmetical functions; the surveys
of Lutz Lucht and Karl-Heinz Indlekofer discuss, among other topics, questions about
inversionofmultiplicativefunctions,resp.integrationtheoryforarithmeticalfunctions.
WolfgangSchwarzhasalwaysbeeninterestedinestablishinginternationalcon-
tacts.Hepursuedthisobjectiveasoneofthemainorganizersofthenumbertheory
meetings at Oberwolfach for two decades from 1974 until 1994 (together with
x Wenneraberstudiert,müssenihmbedeutendeidealeZielevorschweben
Fig.3 Wolfgang Schwarz and his first doctoral student Karl-Heinz Indlekofer discussing the
divisorfunctionin1993.Publishedwithkindpermissionof©Karl-HeinzIndlekofer,1993
Hans-Egon Richert and Eduard Wirsing). The success and worldwide reputation
of these workshops in the Black Forest emphasize the mathematical significance
ofbothWolfgangSchwarzandthegroupofGermannumbertheoristsatthistime.
Anothernotableactivity of Schwarz was the exchangeprogramwith the groupof
probabilistsandnumbertheoristsaroundJonasKubiliusatVilniusUniversity(even
beforeLithuania’sindependencein1990).
InparticulartheseLithuaniancontactsmatchedSchwarz’sresearchinterestsonprobabilis-
ticnumber theory and applications toarithmetical functions, wellreflected inthe article
byEugenijusManstavicˇius.However,questionsaroundtheRiemannzeta-functionandits
relativesshouldbementionedhereaswell.Toillustratethislineofresearchonemaylist
thecontributionsofRamu¯nasGarunkštis&JustasKalpokas,andAntanasLaurincˇikas.The
articleby HelmutMaier& Michael Rassias falls intothiscategory, too.Another related
paperbyNicolaOswald&JörnSteudingtracesthisdirectionofinvestigationintheresearch
areaofthefamousmathematicianAdolfHurwitzwhowasborninHildesheim.5Thisarticle
onthehistoryofnumbertheoryreflectsafurtherresearchinterestofWolfgangSchwarz.
After WolfgangSchwarz’s retirementin 2002,the history of mathematics and,
inparticular,thedevelopmentofnumbertheory,whichhehadalwaysconsideredto
playanimportantroleinmathematicalthinkingandeducation,becamethecentral
topicofhismathematicalactivities.His extraordinaryknowledgeofmathematical
andhistoricaldetailsoftenledtorelevantandhelpfulremarksatconferences,more
than once opening up a new point of view on a discussed unsolved problem or
5Thetownthathostedthe2014E`AZ conferencewherethefirststepsforthismemorialvolume
wereinitiated.
Wenneraberstudiert,müssenihmbedeutendeidealeZielevorschweben xi
Fig.4 WolfgangSchwarzplayingthepiano.Publishedwithkindpermissionof©DorisSchwarz,
2013
a presented result. Wolfgang Schwarz’s other passion was classical music, which
took an even larger role in his life as he was getting older. He played the piano
to an almostprofessionallevel, and he composedhis own pieces of music, which
regularlyhavebeenperformedbymembersofhisfamilyandalsobystudentsofthe
HochschulefürMusikundDarstellendeKunstFrankfurt(forexample,tocelebrate
his75thbirthday).
Fig.5 Eva Schwarz and Valentin Blomer playing one of Wolfgang Schwarz’s pieces at the
memorial colloquium during the 2014 E`AZ conference. Published with kind permission of
©IsaLange,2014