The theory of numbers 数论

Support text for a first course in number theory features the use of algebraic methods for studying arithmetic functions. Subjects covered include the Erd?s-Selberg proof of the Prime Number Theorem, an introduction to algebraic and geometric number theory—the former by studying Gaussian and Jacobian integers, the latter through geometric methods in proving the Quadratic Reciprocity Law and in proofs of certain asymptotic formulas for summatory functions

CHAPTER 1 FUNDAMENTAL CONCEPTS
　1.Divisibility
　2.The gcd and the Icm
　3.The Euclidean Algorithm
　4.The Fundamental Theorem
CHAPTER 2　ARITHMETIC FUNCTIONS
　5. The Semigroup of Arithmetic Functions
　6.The Group of Units in
　7.The Subgroup of Multiplicative Functions
　8.The Mobius Function and Inversion Formulas
　9.The Sigma Functions
　10.The Euler Function
CHAPTER 3　CONGRUENCES AND RESIDUES
　11.Complete Residue Systems
　12.Linear Congruences
　13.Reduced Residue Systems
　14.Ramanujan's Trigonometric Sum
　15.Wilson's Theorem
　16.Primitive Roots
　17.Quadratic Residues
　18.Congruences with Composite Moduli
CHAPTER 4　SUMMATORY FUNCTIONS
　19.Introduction
　20.The Euler-McLaurin Sum Formula
　21.Order of Magnitude of r(n)
　22.Order of Magnitude of a(n)
　23.Sums Involving the M6bius Function
　24. Squarefree Integers
CHAPTER 5 SUMS OF SQUARES
　25.Sums of Four Squares
　26.Sums of Two Squares
　27.Number of Representations
　　28.The Gaussian Integers
　29.Proof of Theorem 27.1
　30.Restatement of Theorem 27.1
CHAPTER 6 CONTINUED FRACTIONS, FAREY　SEQUENCES, THE PELL EQUATIOr
　31.Finite Continued Fractions
　32.Infinite Simple Continued Fractions
　33.Farey Sequences
　34.The Pell Equation
　35.Rational Approximations of Reals
CHAPTER 7 THE EQUATION
　36.Pythagorean Triples
　37.The Equation
　38.Arithmetic in
　39. The Equation
CHAPTER 8 THE PRIME NUMBER THEOREM
　40.Introductory Remarks
　41.Preliminary Results
　42.The Function (x)
　43.A Fundamental Inequality
　44.The Behavior of r(x)/x
　45.The Prime Number Theorem and Related
Results
CHAPTER 9 GEOMETRY OF NUMBERS
　46.Preliminaries
　47.Convex Symmetric Distance Functions
　48.The Theorems of Minkowski
　49.Applications to Farey Sequences and Continued　Fractions
Solutions for Selected Exercises
Bibliography
Index