Problems in Analytic Number Theory

1. Front Matter

   Pages i-xvi

   PDF

2. Problems

   1. Front Matter

      Pages 1-1

      PDF

   2. Arithmetic Functions

      • M. Ram Murty

      Pages 3-15

   3. Primes in Arithmetic Progressions

      • M. Ram Murty

      Pages 17-33

   4. The Prime Number Theorem

      • M. Ram Murty

      Pages 35-51

   5. The Method of Contour Integration

      • M. Ram Murty

      Pages 53-68

   6. Functional Equations

      • M. Ram Murty

      Pages 69-83

   7. Hadamard Products

      • M. Ram Murty

      Pages 85-99

   8. Explicit Formulas

      • M. Ram Murty

      Pages 101-113

   9. The Selberg Class

      • M. Ram Murty

      Pages 115-126

   10. Sieve Methods

       • M. Ram Murty

       Pages 127-145

   11. p-adic Methods

       • M. Ram Murty

       Pages 147-170

3. Solutions

   1. Front Matter

      Pages 171-171

      PDF

   2. Arithmetic Functions

      • M. Ram Murty

      Pages 173-210

   3. Primes in Arithmetic Progressions

      • M. Ram Murty

      Pages 211-245

   4. The Prime Number Theorem

      • M. Ram Murty

      Pages 247-278

   5. The Method of Contour Integration

      • M. Ram Murty

      Pages 279-302

   6. Functional Equations

      • M. Ram Murty

      Pages 303-330

   7. Hadamard Products

      • M. Ram Murty

      Pages 331-356

   8. Explicit Formulas

      • M. Ram Murty

      Pages 357-375

"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).

• Analytic number theory
• Prime
• Prime number
• number theory
• p-adic analytic methods

M.R. Murty

Problems in Analytic Number Theory

"The reviewer strongly approves of the problem-based approach to learning, and recommends this book to any student of analytic number theory."

—MATHEMATICAL REVIEWS

From the reviews of the second edition:

"The second edition of the book has eleven chapters … . the book can be used both as a problem book (as its title shows) and also as a textbook (as the series in which the book is published shows). … is ideal as a text for a first course in analytic number theory, either at the senior undergraduate or the graduate level. … I believe that this book will be very useful for students, researchers and professors. It is well written … ." (Mehdi Hassani, MathDL, April, 2008)

• Department of Mathematics, Queen’s University, Kingston, Canada

  M. Ram Murty

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