historical introduction
chapter 1 the fundamental theorem of arithmetic
1.1 introduction
1.2 divisibility
1.3 greatest common divisor
1.4 prime numbers
1.5 the fundamental theorem of arithmetic
1.6 the series of reciprocals of the primes
1.7 the euclidean algorithm
1.8 the greatest common divisor of more than two numbers
exercises for chapter !
chapter 2 arithmetical functions and dirichlet multiplication
2.1 introduction
2.2 the mebius function mn
2.3 the euler totient function 0n
2.4 a relation connecting 0 and it
2.5 a product formula for n
2.6 the dirichlet product of arithmetical functions
2.7 dirichlet inverses and the mebius inversion formula
2.8 the mangoidt function an
2.9 multiplicativefunctions
2.10 multiplicative functions and dirichlet multiplication
2.11 the inverse of a completely multiplicative function
2.12 liouville''s function ..
2.13 the divisor functions a,n
2.14 generalized convolutions
2.15 formal power series
2.16 the bell series of an arithmetical function
2.17 bell series and dirichlet multiplication
2.18 derivatives of arithmetical functions
2.19 the selberg identity
exercises for chapter 2
chapter 3 averages of arithmetical functions
3.1 introduction
3.2 the big oh notation. asymptotic equality of functions
3.3 euler''s summation formula
3.4 some elementary asymptotic formulas
3.5 the average order old{n}
3.6 the average order of the divisor functions a,n
3.7 the average order ofn
3.8 an application to the distribution of lattice points visible
from the origin
3.9 the average order of un and of an
3.10 the partial sums ora dirichlet product
3.11 applications to #n and an
3.12 another identity for the partial sums of a dirichlet
product
exercises for chapter 3
chapter 4 some elementary theorems on the distribution of
prime
numbers
4.1 introduction
4.2 chebyshev''s functions x and ,9x
4.3 relations connecting x and rix
4.4 some equivalent forms of the prime number theorem
4.5 inequalities for rin and pn
4.6 shapiro''s tauberian theorem
4.7 applications of shapiro''s theorem
4.8 an asymptotic formula for the partial sums σpsx ip
4.9 the partial sums of the m6bius function
4.10 brief sketch of an elementary proof of the prime number
theorem
4.11 selberg''s asymptotic formula
exercises for chapter 4 lot
chapter 5 congruences
5.1 definition and basic properties of congruences
5.2 residue classes and complete residue systems
5.3 linear congruences
5.4 reduced residue systems and the euler-fermat theorem il
5.5 polynomial congruences modulo p. lagrange''s theorem
5.6 applications of lagrange''s theorem
5.7 simultaneous linear congruences. the chinese remainder theorem
l !
5.8 applications of the chinese remainder theorem il
5.9 polynomial congruences with prime power moduli
5.10 the principle of cross-classification
5.11 a decomposition property of reduced residue systems
exercises for chapter 5
chapter 6 finite abelian groups and their characters
6.1 definitions
6.2 examples of groups and subgroups
6.3 elementary properties of groups
6.4 construction of subgroups
6.5 characters of finite abelian groups
6.6 the character group
6.7 the orthogonality relations for characters
6.8 dirichlet characters
6.9 sums involving dirichlet characters
6.10 the nonvanishing of li, x for real nonprincipal x l#l
exercises for chapter 6
chapter 7 dirichlet''s theorem on primes in arithmetic
progressions
7.1 introduction
7.2 dirichlet''s theorem for primes of the form 4n - i and 4n +
i
7.3 the plan of the proof of dirichlet''s theorem
7.4 proof of lemma 7.4
7.5 proof of lemma 7.5
7.6 proof of lemma 7.6
7.7 proof of lemma 7.8
7.8 proof of lemma 7.7
7.9 distribution of primes in arithmetic progressions
exercises for chapter 7
chapter 8 periodic arithmetical functions and gauss sums
8.1 functions periodic modulo k
8.2 existence of finite fourier series for periodic arithmetical
functions
8.3 ramanujan''s sum and generalizations
8.4 multiplicative properties of the sums skn
8.5 gauss sums associated with dirichlet characters
8.6 dirichlet characters with nonvanishing gauss sums
8.7 induced moduli and primitive characters
8.8 further properties of induced moduli
8.9 the conductor of a character
8.10 primitive characters and separable gauss sums
8.11 the finite fourier series of the dirichlet characters
8.12 p61ya''s inequality for the partial sums of primitive
characters
exercises for chapter 8
chapter 9 quadratic residues and the quadratic reciprocity
law
9.1 quadratic residues
9.2 legendre''s symbol and its properties
9.3 evaluation of- lip and 2]p
9.4 gauss'' lemma
9.5 the quadratic reciprocity law
9.6 applications of the reciprocity law
9.7 the jacobi symbol
9.8 applications to diophantine equations
9.9 gauss sums and the quadratic reciprocity law
9.10 the reciprocity law for quadratic gauss sums
9.11 another proof of the quadratic reciprocity law
exercisesfor chapter 9
chapter 10 primitive roots
10.1 the exponent ora number mod m. primitive roots
10.2 primitive roots and reduced residue systems
10.3 the nonexistence of primitive roots mod 2'' for a ] 3
10.4 the existence of primitive roots mod p for odd primes p
10.5 primitive roots and quadratic residues
10.6 the existence of primitive roots mod p
10.7 the existence of primitive roots mod 2p
10.8 the nonexistence of primitive roots in the remaining
cases
10.9 the number of primitive roots mod m
10.10 the index calculus
10.11 primitive roots and dirichlet characters
10.12 real-valued dirichlet characters mod p
10.13 primitive dirichlet characters mod p
exercises for chapter 10
chapter 11 dirichlet series and euler products
11.1 introduction
11.2 the half-plane of absolute convergence of a dirichlet
series
11.3 the function defined by a dirichlet series
11.4 multiplication of dirichlet series
11.5 euler products
11.6 the half-plane of convergence of a dirichlet series
11.7 analytic properties of dirichlet series
11.8 dirichlet series with nonnegative coefficients
11.9 dirichlet series expressed as exponentials of dirichlet
series
11.10 mean value formulas for dirichlet series
11.11 an integral formula for the coefficients of a dirichlet
series
11.12 an integral formula for the partial sums ora dirichlet
series
exercises for chapter ii
chapter 12 the functions ζs and ls, x
12.1 introduction
12.2 properties of the gamma function
12.3 lntegrai representation for the hurwitz zeta function
12.4 a contour integral representation for the hurwitz zeta
function
12.5 the analytic continuation of the hurwitz zeta function
12.6 analytic continuation of ζs and ls, x
12.7 hurwitz''s formula for ζs, a
12.8 the functional equation for the riemann zeta function
12.9 a functional equation for the hurwitz zeta function
12.10 the functional equation for l-functions
12.11 evaluation of ζ-n, a
12.12 properties of bernoulli numbers and bernoulli
polynomials
12.13 formulas for l0, z
12.14 approximation of ζs, a by finite sums
12.15 inequalities for iζs, al
12.16 inequalities for iζsl and ils, xl
exercises for chapter 12
chapter 13 analytic proof of the prime number theorem
13.1 theplan of the proof
13.2 lemmas
13.3 a contour integral representation for ψxx2
13.4 upper bounds for ┃ζs┃and iζ''s[ near the line a =1
13.5 the nonvanishing of ζs on the line a =1
13.6 inequalities for ┃1ζs and ┃ζ''sζs┃
13.7 completion of the proof of the prime number theorem
13.8 zero-free regions for ζs
13.9 the riemann hypothesis
13.10 application to the divisor functi6n
13.11 application to euler''s totient
13.12 extension of pe1ya''s inequality for character sums
exercises for chapter 13
chapter 14 partitions
14.1 introduction
14.2 geometric representation of partitions
14.3 generating functions for partitions
14.4 euler''s pentagonal-number theorem
14.5 combinatorial proof of euler''s pentagonal-number
theorem
14.6 euler''s recursion formula for pn
14.7 an upper bound for pn
14.8 jacobi''s triple product identity
14.9 consequences of jacobi''s identity
14.10 logarithmic differentiation of generating functions
14.11 the partition identities of ramanujan
exercises for chapter 14
bibliography
index of special symbols
index
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