Arithmetic Tales

1. Front Matter

   Pages i-xix

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2. Basic Tools

   • Olivier Bordellès

   Pages 1-18

3. Linear Diophantine Equations

   • Olivier Bordellès

   Pages 19-43

4. Prime Numbers

   • Olivier Bordellès

   Pages 45-157

5. Arithmetic Functions

   • Olivier Bordellès

   Pages 159-353

6. Lattice Points

   • Olivier Bordellès

   Pages 355-409

7. Exponential Sums

   • Olivier Bordellès

   Pages 411-515

8. Algebraic Number Fields

   • Olivier Bordellès

   Pages 517-673

9. Back Matter

   Pages 675-782

   PDF

This textbook covers a wide array of topics in analytic and multiplicative number theory, suitable for graduate level courses.

Extensively revised and extended, this Advanced Edition takes a deeper dive into the subject, with the elementary topics of the previous edition making way for a fuller treatment of more advanced topics. The core themes of the distribution of prime numbers, arithmetic functions, lattice points, exponential sums and number fields now contain many more details and additional topics. In addition to covering a range of classical and standard results, some recent work on a variety of topics is discussed in the book, including arithmetic functions of several variables, bounded gaps between prime numbers à la Yitang Zhang, Mordell's method for exponential sums over finite fields, the resonance method for the Riemann zeta function, the Hooley divisor function, and many others. Throughout the book, the emphasis is on explicit results.

Assuming only familiarity with elementary number theory and analysis at an undergraduate level, this textbook provides an accessible gateway to a rich and active area of number theory. With an abundance of new topics and 50% more exercises, all with solutions, it is now an even better guide for independent study.

• arithmetical functions
• multiplicative number theory
• exponential sums
• analytic number theory
• algebraic number theory
• distribution of prime numbers
• lattice points
• exponent pairs
• Dedekind zeta function
• Dirichlet L-function

“Long chapters with many (exciting) subsections. There are a lot of problems to do (and, yes, do them); there are hints and answers in the back. The book is very well-written, doing a lot of deep stuff in detail and continually providing discussions of recent results, open questions, and so on. There are fantastic lists of references after each chapter. ... It’s a wonderful book.” (Michael Berg, MAA Reviews, January 30, 2022)

• 2 allée de la combe, Aiguilhe, France

  Olivier Bordellès

Olivier Bordellès has been an independent researcher in number theory for 15 years, essentially working in multiplicative number theory and the links between algebraic number theory and analytic number theory.

Véronique Bordellès is an English teacher who has translated several books in maths and science. She holds a BA in linguistics from the University of Manchester (UK) and also completed a Master's thesis in linguistics.

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