HistoricalIntroduction
Chapter1
TheFundamentalTheoremofArithmetic
1.1Introduction
1.2Divisibility
1.3Greatestcommondivisor
1.4Primenumbers
1.5Thefundamentaltheoremofarithmetic
1.6Theseriesofreciprocalsoftheprimes
1.7TheEuclideanalgorithm
1.8Thegreatestcommondivisorofmorethantwo,numbers
ExercisesforChapter1
Chapter2
ArithmeticalFunctionsandDirichletMultiplication
2.1Introduction
2.2TheM6biusfunction(n)
2.3TheEulertotientfunction(n)
2.4Arelationconnectingandu
2.5Aproductformulafor(n)
2.6TheDirichletproductofarithmeticalfunctions
2.7DirichletinversesandtheM6biusinversionformula
2.8TheMangoldtfunctionA(n)
2.9Muitiplicativefunctions
2.10MultiplicativefunctionsandDirichletmultiplication
2.11Theinverseofacompletelymultiplicativefunction
2.12Liouville'sfunction)
2.13Thedivisorfunctionsa,(n)
2.14Generalizedconvolutions
2.15Formalpowerseries
2.16TheBellseriesofanarithmeticalfunction
2.17BellseriesandDirichletmultiplication
2.18Derivativesofarithmeticalfunctions
2.19TheSelbergidentity
ExercisesforChapter2
Chapter3
AveragesofArithmeticalFunctions
3.1Introduction
3.2Thebigohnotation.Asymptoticequalityoffunctions
3.3Euler'ssummationformula
3.4Someelementaryasymptoticformulas
3.5Theaverageorderofdin)
3.6Theaverageorderofthedivisorfunctionsa,(n)
3.7Theaverageorderof~0(n)
3.8Anapplicationtothedistributionoflatticepointsvisiblefromtheorigin
3.9Theaverageorderof/4n)andofA(n)
3.10ThepartialsumsofaDirichletproduct
3.11Applicationstopin)andA(n)
3.12AnotheridentityforthepartialgumsofaDirichletproduct
ExercisesforChapter3
Chapter4
SomeElementaryTheoremsontheDistributionofPrime
Numbers
4.1Introduction
4.2Chebyshev'sfunctions(x)and(x)
4.3Relationsconnecting/x)andn(x)
4.4Someequivalentformsoftheprimenumbertheorem
4.5Inequalitiesfor(n)andp,
4.6Shapiro'sTauberiantheorem
4.7ApplicationsofShapiro'stheorem
4.8Anasymptoticformulaforthepartialsums,(I/p)
4.9ThepartialsumsoftheM6biusfunction91
4.10Briefsketchofanelementaryproofoftheprimenumbertheorem
4.11Selbcrg'sasymptoticformula
ExercisesforChapter4
Chapter5
Congruences
5.1Definitionandbasicpropertiesofcongruences
5.2Residueclassesandcompleteresiduesystems
5.3Linearcongruences
Chapter6
FiniteAbelianGroupsandTheirCharacters
Chapter7
Dirichlet'sTheoremonPrimesinArithmeticProgressions
Chapter8
PeriodicArithmeticalFunctionsandGaussSums
Chapter9
QuadraticResiduesandtheQuadraticReciprocityLaw
Chapter10
PrimitiveRoots
Chapter11
DirichletSeriesandEulerProducts
Chapter12
TheFunctions(s)andL(s,x)
Chapter13
AnalyticProofofthePrimeNumberTheorem