Table Of ContentA REVIEW OF THE VON STAUDT CLAUSEN THEOREM
by
TimothySimonCaley
SUBMITTEDINPARTIALFULFILLMENTOFTHE
REQUIREMENTSFORTHEDEGREEOF
MASTEROFSCIENCE
AT
DALHOUSIEUNIVERSITY
HALIFAX,NOVASCOTIA
MARCH2007
(cid:13)c CopyrightbyTimothySimonCaley,2007
DALHOUSIEUNIVERSITY
DEPARTMENTOFMATHEMATICSANDSTATISTICS
The undersigned hereby certify that they have read and recommend to the
Faculty of Graduate Studies for acceptance a thesis entitled (cid:147)A REVIEW OF THE VON
STAUDT CLAUSEN THEOREM(cid:148) by TimothySimonCaley in partial ful(cid:2)llment of the
requirementsforthedegreeofMasterofScience.
Dated: March29,2007
Supervisor:
KarlDilcher
Readers:
KeithJohnson
RobertMilson
ii
DALHOUSIE UNIVERSITY
DATE:March29,2007
AUTHOR: TimothySimonCaley
TITLE: AREVIEWOFTHEVONSTAUDTCLAUSENTHEOREM
DEPARTMENTORSCHOOL: DepartmentofMathematicsandStatistics
DEGREE:M.Sc. CONVOCATION:May YEAR:2007
Permission is herewith granted to Dalhousie University to circulate and to have copied
for non-commercial purposes, at its discretion, the above title upon the request of individuals or
institutions.
SignatureofAuthor
The author reserves other publication rights, and neither the thesis nor extensive extracts
fromitmaybeprintedorotherwisereproducedwithouttheauthor’swrittenpermission.
The author attests that permission has been obtained for the use of any copyrighted
material appearing in the thesis (other than brief excerpts requiring only proper acknowledgement
inscholarlywriting)andthatallsuchuseisclearlyacknowledged.
iii
Table of Contents
ListofFigures vii
ListofAbbreviationsandSymbolsUsed viii
Abstract ix
Acknowledgements x
Chapter1 Introduction 1
Chapter2 BackgroundMaterial 7
2.1 ElementaryPropertiesofBernoulliNumbers . . . . . . . . . . . . . . . . . 7
2.2 RecurrenceRelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 BernoulliPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 TheRiemannZetaFunction . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Euler-MacLaurinSummationFormula . . . . . . . . . . . . . . . . . . . . 16
2.6 AnotherRecurrenceRelation . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 Analyticand p-adicBackground . . . . . . . . . . . . . . . . . . . . . . . 22
2.8 BinomialCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.9 SomeAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.10 StirlingNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.11 CalculusofFiniteDifferences . . . . . . . . . . . . . . . . . . . . . . . . 32
2.12 AnIn(cid:2)niteSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.13 AnExplicitFormulaforBernoulliNumbers . . . . . . . . . . . . . . . . . 37
2.14 HurwitzSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Chapter3 ProofsofthevonStaudt-ClausenTheorem 51
3.1 AnImportantLemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 EquivalenceofStatementsoftheTheorem . . . . . . . . . . . . . . . . . . 52
iv
3.3 ElementaryProofsofthevonStaudt-ClausenTheorem . . . . . . . . . . . 54
3.3.1 vonStaudt’sProof . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 ProofsfromtheExplicitFormula . . . . . . . . . . . . . . . . . . 57
3.3.3 Catalan’sProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.4 TheProofofLucas . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.5 Chowla’sProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3.6 Rado’sProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3.7 AProoffromtheUmbralCalculus . . . . . . . . . . . . . . . . . . 76
3.3.8 Witt’sProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3.9 Washington’sProof . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.3.10 Howard’sProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.4 OtherProofsofthevonStaudt-ClausenTheorem . . . . . . . . . . . . . . 86
3.4.1 Stevens’Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4.2 Prooffrom p-adicIdeas . . . . . . . . . . . . . . . . . . . . . . . 92
3.4.3 ProoffromKummer’sCongruence . . . . . . . . . . . . . . . . . . 93
Chapter4 ApplicationsandConsequences 94
4.1 FurtherCongruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.1.1 Voronoi’sCongruence . . . . . . . . . . . . . . . . . . . . . . . . 94
4.1.2 J.C.AdamsTheorem . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.1.3 Kummer’sCongruence . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2 OtherElementaryApplications . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2.1 FermatPrimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2.2 BinomialCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.2.3 CalculationofBernoulliNumbers . . . . . . . . . . . . . . . . . . 96
4.3 Giuga’sConjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4 FractionalPartsofBernoulliNumbers . . . . . . . . . . . . . . . . . . . . 97
4.5 IrregularPrimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Chapter5 Conclusion 100
v
AppendixA BiographiesofStaudtandClausen 102
A.1 ThomasClausen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A.1.1 Clausen’sPublication . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.2 K.G.C.vonStaudt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.2.1 vonStaudt’sPublication . . . . . . . . . . . . . . . . . . . . . . . 105
Bibliography 109
vi
List of Figures
Figure4.1 F (z)for x = 100000 . . . . . . . . . . . . . . . . . . . . . . . . . 99
x
FigureA.1 ThomasClausen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
FigureA.2 Clausen’spublication . . . . . . . . . . . . . . . . . . . . . . . . . 104
FigureA.3 K.G.C.vonStaudt . . . . . . . . . . . . . . . . . . . . . . . . . . 105
FigureA.4 vonStaudt’spublication,page1 . . . . . . . . . . . . . . . . . . . 106
FigureA.5 vonStaudt’spublication,page2 . . . . . . . . . . . . . . . . . . . 107
FigureA.6 vonStaudt’spublication,page3 . . . . . . . . . . . . . . . . . . . 108
vii
List of Abbreviations and Symbols Used
Re(s)....... realvalueof s
B ....... Bernoullinumber
n
D ....... denominatorofthe nthBernoullinumber
n
B (x)....... nthBernoullipolynomial
n
ζ(s)....... Riemannzetafunction
Z ....... ringof p-integers
p
∗
A ....... groupofunitsof A
χ....... Dirichletcharacter
fχ ....... conductorof χ
L(s,χ)....... L-seriesattachedto χ
Bn,χ ....... generalizedBernoullinumber
S(n,k)....... Stirlingnumberofthesecondkind
⇔....... ifandonlyif
∆....... forwarddifferenceoperator
B....... Hurwitzseries
D....... differentialoperator
(cid:98)x(cid:99)....... greatestinteger≤ x
F ....... Fermatnumber
n
T ....... tangentnumber
n
{x}....... thefractionalpartof x
viii
Abstract
The purpose of this thesis is to give a review of the different proofs of the von Staudt-
Clausen Theorem that appear in the literature. There is a relatively large number of proofs
that each use different ideas. The (cid:2)rst chapter will contain a brief historical discussion of
the von Staudt-Clausen Theorem and its discovery. The second chapter contains all of the
background material that is necessary to understand the various proofs of the von Staudt-
Clausen Theorem. In general, most of this material appears widely in the literature, but
often without proof, which I will supply. The proofs of the von Staudt-Clausen Theorem
willbegiveninthethirdchapter. Foreachproof,Iwilldiscussthetechniquesusedaswell
as any relevant historical background or mistakes in the original, if this is necessary. In
the literature, there are a number of proofs that are similar to one another, and so I have
grouped them together in subsections. I have also tried to determine who (cid:2)rst discovered
eachmethodofproofandwhetherornotanysubsequentproofsthatappearintheliterature
are independent. The fourth chapter will brie(cid:3)y deal with some of the applications and
consequences of the von Staudt-Clausen Theorem. This theorem is useful in many areas,
particularly proving other congruences concerning Bernoulli numbers, but I only include
the most important and interesting results. Finally, in the (cid:2)fth chapter, I will sum up my
(cid:2)ndingsandcomparethedifferentproofs.
ix
Acknowledgements
I would like to thank my supervisor Professor Dilcher for his help, patience, expertise, ad-
vice and (cid:2)nancial support he gave me in the last year, and being an excellent supervisor
in general. I would not have been able to complete my thesis without his dedication. I
would also like to thank my readers Professors Johnson and Milson for taking the time to
read my thesis and for their very helpful corrections and comments. Furthermore, I would
liketothanktwocolleaguesofProfessorDilcher,ProfessorsT.AgohandI.Slavutskiiwho
assisted me with translating and locating obscure papers. I also wish to acknowledge the
(cid:2)nancial support I received from the Faculty of Graduate Studies and the Department of
Mathematics and Statistics. Finally, thank you to my parents and Laura for the uncondi-
tionalloveandsupportyougavemeduringthistime.
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