Problems in Analytic Number Theory

This book gives a problem-solving approach to the difficult subject of analytic number theory. It is primarily aimed at graduate students and senior undergraduates. The goal is to provide a rapid introduction to analytic methods and the ways in which they are used to study the distribution of prime numbers. The book also includes an introduction to p-adic analytic methods. It is ideal for a first course in analytic number theory. The new edition has been completely rewritten, errors have been corrected, and there is a new chapter on equidistribution.

About the first edition:

"...this monograph gives important results and techniques for specific topics, together with many exercises; it is not possible to describe adequately the wealth of material covered in this book."

• Wolfgang Schwarz, Zentralblatt

Texte du rabat

"In order to become proficient in mathematics, or in any subject," writes Andre Weil, "the student must realize that most topics in­ volve only a small number of basic ideas. " After learning these basic concepts and theorems, the student should "drill in routine exercises, by which the necessary reflexes in handling such concepts may be ac­ quired. . . . There can be no real understanding of the basic concepts of a mathematical theory without an ability to use them intelligently and apply them to specific problems. " Weil's insightfulobservation becomes especially important at the graduate and research level. It is the viewpoint of this book. Our goal is to acquaint the student with the methods of analytic number theory as rapidly as possible through examples and exercises. Any landmark theorem opens up a method of attacking other problems. Unless the student is able to sift out from the mass of theory the underlying techniques, his or her understanding will only be academic and not that of a participant in research. The prime number theorem has given rise to the rich Tauberian theory and a general method of Dirichlet series with which one can study the asymptotics of sequences. It has also motivated the development of sieve methods. We focus on this theme in the book. We also touch upon the emerging Selberg theory (in Chapter 8) and p-adic analytic number theory (in Chapter 10).

Contenu

I Problems.- 1 Arithmetic Functions.- 2 Primes in Arithmetic Progressions.- 3 The Prime Number Theorem.- 4 The Method of Contour Integration.- 5 Functional Equations.- 6 Hadamard Products.- 7 Explicit Formulas.- 8 The Selberg Class.- 9 Sieve Methods.- 10 p-adic Methods.- II Solutions.- 1 Arithmetic Functions.- 2 Primes in Arithmetic Progressions.- 3 The Prime Number Theorem.- 4 The Method of Contour Integration.- 5 Functional Equations.- 6 Hadamard Products.- 7 Explicit Formulas.- 8 The Selberg Class.- 9 Sieve Methods.- 10 p-adic Methods.- References.