Introduction

Summary

Through this book we wish to achieve and connect the following three goals:

1. 1) to present some elementary results in number theory;

2. 2) to introduce classical and recent topics on the uniform distribution of infinite sequences and on the discrepancy of finite sequences in several variables;

3. 3) to present a few results in Fourier analysis and use them to prove some of the theorems discussed in the two previous points.

The first part of this book is dedicated to the first goal. The reader will find some topics typically presented in introductory books on number theory: factorization, arithmetic functions and integer points, congruences and cryptography, quadratic reciprocity, and sums of two and four squares. Starting from the first few pages we introduce some simple and captivating findings, such as Chebyshev's theorem and the elementary results for the Gauss circle problem and for the Dirichlet divisor problem, which may lead the reader to a deeper study of number theory, particularly students who are interested in calculus and analysis.

In the second part we start with the uniformly distributed sequences, introduced in 1916 by Weyl and related to the strong law of large numbers and to Kronecker's approximation theorem. Then we introduce the definition of discrepancy, which is the quantitative counterpart of the uniform distribution and has natural applications in the computation of high-dimensional integrals.