Preface Number theory is one of the most fascinating topics in mathematics, and there are various reasons for this. Here are a few: • Several number theory problems can be formulated in simple terms with very little or no background required to understand their state- ments. • It has a rich history that goes back thousands of years when mankind was learning to count (even before learning to write!). • Some of the most famous minds of mathematics (including Pascal, Euler, Gauss, and Riemann, to name only a few) have brought their contributions to the development of number theory. • Like many other areas of science, but perhaps more so with this one, its development suffers from an apparent paradox: giant leaps have been made over time, while some problems remain as of today completely impenetrable, with little or no progress being made. All this explains in part why so many scientists and so many amateurs have worked on famous problems and conjectures in number theory. The long quest for a proof of Fermat’s Last Theorem is only one example. And what about “analytic number theory”? The use of analysis (real or complex) to study number theory problems has brought light and elegance to this field, in particular to the problem of the distribution of prime numbers. Through the centuries, a large variety of tools has been developed to grasp a better understanding of this particular problem. But the year 1896 saw a turning point in the history of number theory. Indeed, that was the year when two mathematicians, Jacques Hadamard and Charles Jean de la Vall´ee Poussin, one French, the other Belgian, independently used complex analysis ix

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