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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.
List of contents
1 Basic Tools.- 2 Linear Diophantine Equations.- 3 Prime Numbers.- 4 Arithmetic Functions.- 5 Lattice Points.- 6 Exponential Sums.- 7 Algebraic Number Fields. Hints and Answers to Exercises.- Index.
About the author
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.
Report
"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)
