Number Theory

The following sections are included:

• FOREWORD

• CONTENTS

• NOTATION

The following sections are included:

• Divisibility. Prime numbers

• Linear and quadratic Diophantine equations

• Congruences

• Quadratic congruences

• An application of trigonometrical sums in the theory of numbers

The following sections are included:

• Fundamental properties

• Additive and multiplicative functions

• The natural density. Average values of arithmetical functions

The fundamental purposes of this chapter are to prove the so-called prime number theorem giving an asymptotic formula for the number π(x) of primes not exceeding x and to prove Dirichlet’s theorem on primes in arithmetic progressions. We shall begin by proving some elementary results, and then discuss analytic methods which will be used to prove the fundamental theorems of this chapter.

The following sections are included:

• Eratosthenes’ sieve

• Selberg’s sieve

• The large sieve

The following sections are included:

• Convex sets

• Minkowski’s theorem

• Lattice points in plane regions

The following sections are included:

• Schnirelman’s density

• The Waring-Hilbert theorem

• Other additive problems

The following sections are included:

• The Turán - Kubilius inequality

• The Erdös - Kac theorem

• Asymptotic distribution functions

In this chapter we shall present fundamental results concerning the approximation of real numbers by rational numbers and we shall also speak about problems of distribution of sequences of numbers in the interval [0, 1).

The following sections are included:

• Algebraic numbers and algebraic integers

• Ideals in the rings of algebraic integers

• p-adic numbers

The following sections are included:

• LITERATURE CITED

• INDEX