Multiplicative Number Theory I
Classical Theory

A 2006 text based on courses taught successfully over many years at Michigan, Imperial College and Pennsylvania State.

Hugh L. Montgomery (Author), Robert C. Vaughan (Author)

9781107405820, Cambridge University Press

Paperback / softback, published 26 July 2012

572 pages
22.9 x 15.2 x 3.2 cm, 0.83 kg

'The text is very well written and accessible to students. On many occasions the authors explicitly describe basic methods known to everyone working in the field, but too often skipped in textbooks. This book may well become the standard introduction to analytic number theory.' Zentralblatt MATH

Prime numbers are the multiplicative building blocks of natural numbers. Understanding their overall influence and especially their distribution gives rise to central questions in mathematics and physics. In particular their finer distribution is closely connected with the Riemann hypothesis, the most important unsolved problem in the mathematical world. Assuming only subjects covered in a standard degree in mathematics, the authors comprehensively cover all the topics met in first courses on multiplicative number theory and the distribution of prime numbers. They bring their extensive and distinguished research expertise to bear in preparing the student for intelligent reading of the more advanced research literature. This 2006 text, which is based on courses taught successfully over many years at Michigan, Imperial College and Pennsylvania State, is enriched by comprehensive historical notes and references as well as over 500 exercises.

Preface
Notation
1. Dirichlet series-I
2. The elementary theory of arithmetic functions
3. Principles and first examples of sieve methods
4. Primes in arithmetic progressions-I
5. Dirichlet series-II
6. The prime number theorem
7. Applications of the prime number theorem
8. Further discussion of the prime number theorem
9. Primitive characters and Gauss sums
10. Analytic properties of the zeta function and L-functions
11. Primes in arithmetic progressions-II
12. Explicit formulae
13. Conditional estimates
14. Zeros
15. Oscillations of error terms
Appendix A. The Riemann-Stieltjes integral
Appendix B. Bernoulli numbers and the Euler-MacLaurin summation formula
Appendix C. The gamma function
Appendix D. Topics in harmonic analysis.

Subject Areas: Number theory [PBH]