The Theory of Numbers An Introduction

by Gioia, Anthony A.

• ISBN13: 9780486414492
• ISBN10: 0486414493
• Format: Paperback
• Copyright: 2001-03-28
• Publisher: Dover Publications

Summary

Support text for a first course in number theory features the use of algebraic methods for studying arithmetic functions. Subjects covered include the Erdouml;s-Selberg proof of the Prime Number Theorem, an introduction to algebraic and geometric number theorythe 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.

Table of Contents

Preface	vii
Fundamental Concepts
Divisibility	1	(3)
The god and the lcm	4	(3)
The Euclidean Algorithm	7	(2)
The Fundamental Theorem	9	(4)
Arithmetic Functions
The Semigroup A of Arithmetic Functions	13	(3)
The Group of Units in A	16	(2)
The Subgroup of Multiplicative Functions	18	(2)
The Mobius Function and Inversion Formulas	20	(3)
The Sigma Functions	23	(4)
The Euler &Thetas;-Function	27	(4)
Congruences and Residues
Complete Residue Systems	31	(4)
Linear Congruences	35	(4)
Reduced Residue Systems	39	(2)
Ramanujan's Trigonometric Sum	41	(4)
Wilson's Theorem	45	(2)
Primitive Roots	47	(5)
Quadratic Residues	52	(7)
Congruences with Composite Moduli	59	(4)
Summatory Functions
Introduction	63	(7)
The Euler-McLaurin Sum Formula	70	(3)
Order of Magnitude of τ(n)	73	(3)
Order of Magnitude of σ(n)	76	(2)
Sums Involving the Mobius Function	78	(4)
Squarefree Integers	82	(2)
Sums of Squares
Sums of Four Squares	84	(3)
Sums of Two Squares	87	(3)
Number of Representations	90	(1)
The Gaussian Integers	90	(5)
Proof of Theorem 27.1	95	(1)
Restatement of Theorem 27.1	96	(2)
Continued Fractions, Farey Sequences, The Pell Equation
Finite Continued Fractions	98	(5)
Infinite Simple Continued Fractions	103	(4)
Farey Sequences	107	(3)
The Pell Equation	110	(4)
Rational Approximations of Reals	114	(6)
The Equation xn + yn = zn, n ≤ 4
Pythagorean Triples	120	(2)
The Equation x4 + y4 = z4	122	(1)
Arithmetic in K (√-3)	123	(3)
The Equation x3 + y3 = z3	126	(3)
The Prime Number Theorem
Introductory Remarks	129	(1)
Preliminary Results	130	(4)
The Function ψ(x)	134	(6)
A Fundamental Inequality	140	(4)
The Behavior of r(x)/x	144	(9)
The Prime Number Theorem and Related Results	153	(7)
Geometry of Numbers
Preliminaries	160	(4)
Convex Symmetric Distance Functions	164	(6)
The Theorems of Minkowski	170	(4)
Applications to Farey Sequences and Continued Fractions	174	(7)
Solutions for Selected Exercises	181	(22)
Bibliography	203	(2)
Index	205