Non-vanishing of L-Functions and Applications

205 Pages·1997·2.339 MB·English

by  M. Ram Murty, V. Kumar Murty (auth.)

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M. Ram Murty
V. Kumar Mu rty
Non-vanishing
of L-Functions
and Applications
Reprint of the 1997 Edition
M. Ram Murty V. Kumar Murty
Department of Mathematics and Statistics Department of Mathematics
Jeffery Hall, Queen’s University University of Toronto
Kingston, ON, K7L 3N6 40, St. George Street
Canada Toronto, ON M5S 2E4
Canada
ISBN978-3-0348-0273-4 e-ISBN978-3-0348-0274-1
DOI10.1007/978-3-0348-0274-1
Springer Basel Dordrecht Heidelberg London New York
Library of Congress Control Number: 2011941445
MathematicsSubjectClassification (2010): 11Mxx, 11M41, 11G40, 11R52, 11R42
© Springer Basel AG 1997
Reprint of the 1st edition 1997 by Birkhäuser Verlag, Switzerland
Original ly publishe d as volum e 157 i n the P ro gress in Ma thematics series
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Fernando Sunyer i Balaguer 1912–1967
∗ ∗ ∗
This book has been awarded the Ferran Sunyer i Balaguer 1996 prize.
Each year, in honor of the memory of Ferran Sunyer i Balaguer, the Institut
d’Estudis Catalans awards an international research prize for a mathematical
monographofexpositorynature.Theprize-winningmonographsarepublished
in this series. Details about the prize can be found at
http://www.iec.es/fsbprang.htm
Previous winners include
– Alexander Lubotzky
Discrete Groups, Expanding
Graphs and Invariant Measures
(vol. 125)
– Klaus Schmidt
Dynamical Systems of Algebraic
Origin (vol. 128)
Fernando Sunyer i Balaguer 1912–1967
Born in Figueras (Gerona) with an almost fully incapacitating physical disability,
Fernando Sunyer i Balaguer was confined for all his life to a wheelchair he could
not move himself, and was thus constantly dependent on the care of others. His
father died when Don Fernando was two years old, leaving his mother, Don˜a An-
gelaBalaguer,alonewiththeheavyburdenofnursingherson.Theysubsequently
moved in with Fernando’s maternal grandmother and his cousins Maria, Ange-
les, and Fernando. Later, this exemplary family, which provided the environment
of overflowing kindness in which our famous mathematician grew up, moved to
Barcelona.
As the physician thought it advisable to keep the sickly boy away from all
sorts of possible strain, such as education and teachers, Fernando was left with
theoptiontolearneitherbyhimselforthroughhismother’slessonswhich,thanks
to her love and understanding, were considered harmless to his health. Without a
doubt, this education was strongly influenced by his living together with cousins
who were to him much more than cousins for all his life. After a period of in-
tense reading, arousing a first interest in astronomy and physics, his passion for
mathematics emerged and dominated his further life.
In 1938, he communicated his first results to Prof. J. Hadamard of the
Academy of Sciences in Paris, who published one of his papers in the Academy’s
“ComptesRendus”andencouragedhimtoproceedinhisselected courseof inves-
tigation. From this moment, Fernando Sunyer i Balaguer maintained a constant
interchangewiththeFrenchanalyticalschool,inparticularwithMandelbrojtand
his students. In the following years, his results were published regularly. The lim-
itedspaceheredoesnot,unfortunately,allowforacriticalanalysisofhisscientific
achievements. In the mathematical community his work, for which he attained
international recognition, is well known.
Don Fernando’s physical handicap did not allow him to write down any of
hispapersbyhimself.Hedictatedthemtohismotheruntilherdeathin1955,and
when,afteraperiodofgriefanddesperation,heresumedresearchwithnewvigor,
his cousins took care of the writing. His working power, paired with exceptional
talents, produced a number of results which were eventually recognized for their
high scientific value and for which he was awarded various prizes. These honours
notwithstanding,itwasdifficultforhimtoreach thesocialandprofessionalposi-
tion correspondingto his scientific achievements. At times, hiseconomic situation
was notthe most comfortableeither. It wasn’t until the9th of December 1967, 18
dayspriorhisdeath,thathisconfirmationasascientificmemberwasmadepublic
bytheDivisio´ndeCiencias,M´edicasydeNaturalezaoftheCouncil.Furthermore,
he was elected only as “de entrada”, in contrast to class membership.
Due to his physical constraints, the academic degrees for his official studies
were granted rather belatedly. By the time he was given the Bachelor degree, he
hadalreadybeenhonouredbyseveraluniversities!In1960hefinishedhisMaster’s
Fernando Sunyer i Balaguer 1912–1967 vii
degree and was awarded the doctorate after the requisite period of two years as a
student.Althoughhehadbeenapart-timeemployeeoftheMathematicalSeminar
since1948,hewasnotallowedto becomeafullmember ofthescientificstaff until
1962. This despite his actually heading the department rather than just being a
staff member.
Hisown papersregularlyappearedin the journalsof the BarcelonaSeminar,
Collectanea Mathematica, to which he was also an eminent reviewer and advisor.
Onseveraloccasions,hewasconsultedbytheProceedingsoftheAmericanSociety
of Mathematics as an advisor. He always participated in and supported guest
lecturesinBarcelona,manyofthemhavingbeenpreparedorpromotedbyhim.On
the occasion of a conference in 1966, H. Mascart of Toulouse publicly pronounced
his feeling of beeing honoured by the presence of M. Sunyer Balaguer, “the first,
by far, of Spanish mathematicians”.
At all times, Sunyer Balaguer felt a strong attachment to the scientific ac-
tivities of his country and modestly accepted the limitations resulting from his
attitude, resisting several calls from abroad, in particular from France and some
institutions in the USA. In 1963 he was contracted by the US Navy, and in the
following years he earned much respect for the results of his investigations. “His
value to the prestige of the Spanish scientific community was outstanding and his
workinmathematicsofasteadyexcellencethatmakeshislossdifficulttoaccept”
(letter of condolence from T.B. Owen, Rear Admiral of the US Navy).
Twice,SunyerBalaguerwasapproachedbyyoungforeignstudentswhowant-
edtowritetheirthesisunderhissupervision,buthehadtodeclinebecausehewas
unabletoraisethenecessaryscholarshipmoney.Manytimesherevieweddoctoral
theses for Indian universities, on one occasion as the president of a distinguished
international board. The circumstances under which Sunyer attained his scientific
achievements, also testify to his remarkable human qualities. Indeed, his manner
was friendly and his way of conversation reflected his gift for friendship as well
as enjoyment of life and work which went far beyond a mere acceptance of the
situation into which he had been born. His opinions were as firm as they were
cautious, and at the same time he had a deep respect for the opinion and work
of others. Though modest by nature, he achieved due credit for his work, but his
petitionswerefreeofanytraceofexaggerationorundueself-importance.Themost
surprisingofhisqualitieswas,aboveall,hisabsolutelackofpreoccupationwithhis
physical condition, which can largely be ascribed to the sensible education given
by his mother and can be seen as an indication of the integration of the disabled
into our society.
OnDecember27,1967,stillfullyactive,FerranSunyerBalaguerunexpectedly
passed away. The memory of his remarkable personality is a constant source of
stimulation for our own efforts.
TranslatedfromJuanAug´e:FernandoSunyerBalaguer.Gazeta Matematica,
1.a Serie – Tomo XX – Nums. 3 y 4, 1968, where a complete bibliography can be
found.
Satyam Jnanam Anantam Brahma
Table of Contents
Preface .................................................................. xi
Introduction ............................................................. 1
Chapter 1 The Prime Number Theorem and Generalizations
§1 The Prime Number Theorem ........................................ 5
§2 Primes in Arithmetic Progression .................................... 15
§3 Dedekind’s zeta function ............................................ 19
§4 Hecke’sL-functions .................................................. 21
Chapter 2 Artin L-Functions
§1 Group-theoretic background ......................................... 25
§2 Definition and basic properties of Artin L-functions .................. 27
§3 The Aramata-Brauer Theorem....................................... 30
§4 Dedekind’s conjecture in the non-Galois case ........................ 32
§5 Zeros and poles of Artin L-functions ................................. 35
§6 Low order zeros of Dedekind zeta functions .......................... 37
§7 Chebotarev density theorem ......................................... 41
§8 Consequences of Artin’s conjecture .................................. 46
§9 The least prime in a conjugacy class ................................. 52
Chapter 3 Equidistribution and L-Functions
§1 Compact groups and Haar measures ................................. 65
§2 Weyl’s criterion for equidistribution ................................. 66
§3 L-functions on G .................................................... 67
§4 Deligne’s Prime Number Theorem ................................... 68
ix