x Preface to prove what we now call the Prime Number Theorem, namely the fact that “π(x) is asymptotic to x/ log x” as x tends to infinity, where π(x) stands for the number of prime numbers not exceeding x. This event marked the birth of analytic number theory. At first one might wonder how analysis can be of any help in solving problems from number theory, which are after all related to the study of positive integers. Indeed, while integers “live” in a discrete world, analysis “lives” in a continuous one. This duality goes back to Euler, who had observed that there was a connection between an infinite product running over the set of all prime numbers and an infinite series which converges or diverges depending on the value of real variable s, a connection described by the relation 1 1 2s −1 1 1 3s −1 1 1 5s −1 · · · = 1 1s + 1 2s + 1 3s + 1 4s + 1 5s + · · · , which holds in particular for all real s 1. Approximately one century later, Riemann studied this identity for complex values of s by carefully exhibiting the analytic properties of the now famous Riemann Zeta Function ζ(s) = n=1 1 ns (Re(s) 1). By extending this function to the entire complex plane, he used it to establish in 1859 a somewhat exceptional but nevertheless incomplete proof of the Prime Number Theorem. Thirty-seven years later, Hadamard and de la Vall´ ee Poussin managed to complete the proof initiated by Riemann. The methods put forward by Riemann and many other 20th-century mathematicians have helped us gain a better understanding of the distri- bution of prime numbers and a clearer picture of the complexity of the multiplicative structure of the integers or, using a stylistic device, a better comprehension of the anatomy of integers. In this book, we provide an introduction to analytic number theory. The choice of the subtitle “Exploring the Anatomy of Integers” was coined at a CRM workshop held at Universit´ e de Montr´ eal in March 2006 which the two of us, along with Andrew Granville, organized. For the workshop as well as for this book, the terminology “anatomy of integers” is appropriate as it describes the area of multiplicative number theory that relates to the size and distribution of the prime factors of positive integers and of various families of integers of particular interest. Besides the proof of the Prime Number Theorem, our choice of subjects for this book is very subjective but nevertheless legitimate. Hence, several chapters are devoted to the study of arithmetic functions, in particular those
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