Multiplicative Number Theory

1. Front Matter

   Pages i-xiii

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2. Primes in Arithmetic Progression

   • Harold Davenport

   Pages 1-11

3. Gauss’ Sum

   • Harold Davenport

   Pages 12-16

4. Cyclotomy

   • Harold Davenport

   Pages 17-26

5. Primes in Arithmetic Progression: The General Modulus

   • Harold Davenport

   Pages 27-34

6. Primitive Characters

   • Harold Davenport

   Pages 35-42

7. Dirichlet’s Class Number Formula

   • Harold Davenport

   Pages 43-53

8. The Distribution of the Primes

   • Harold Davenport

   Pages 54-58

9. Riemann’s Memoir

   • Harold Davenport

   Pages 59-64

10. The Functional Equation of the L Functions

    • Harold Davenport

    Pages 65-72

11. Properties of the Γ Function

    • Harold Davenport

    Pages 73-73

12. Integral Functions of Order 1

    • Harold Davenport

    Pages 74-78

13. The Infinite Products for ξ(s) and ξ(s, χ)

    • Harold Davenport

    Pages 79-83

14. A Zero-Free Region for ζ(s)

    • Harold Davenport

    Pages 84-87

15. Zero-Free Regions for L(s, χ)

    • Harold Davenport

    Pages 88-96

16. The Number N(T)

    • Harold Davenport

    Pages 97-100

17. The Number N(T, χ)

    • Harold Davenport

    Pages 101-103

18. The Explicit Formula for Ψ(x)

    • Harold Davenport

    Pages 104-110

19. The Prime Number Theorem

    • Harold Davenport

    Pages 111-114

20. The Explicit Formula for ψ(x, χ)

    • Harold Davenport

    Pages 115-120

Although it was in print for a short time only, the original edition of Multiplicative Number Theory had a major impact on research and on young mathematicians. By giving a connected account of the large sieve and Bombieri's theorem, Professor Davenport made accessible an important body of new discoveries. With this stimula­ tion, such great progress was made that our current understanding of these topics extends well beyond what was known in 1966. As the main results can now be proved much more easily. I made the radical decision to rewrite §§23-29 completely for the second edition. In making these alterations I have tried to preserve the tone and spirit of the original. Rather than derive Bombieri's theorem from a zero density estimate tor L timctions, as Davenport did, I have chosen to present Vaughan'S elementary proof of Bombieri's theorem. This approach depends on Vaughan's simplified version of Vinogradov's method for estimating sums over prime numbers (see §24). Vinogradov devised his method in order to estimate the sum LPH e(prx); to maintain the historical perspective I have inserted (in §§25, 26) a discussion of this exponential sum and its application to sums of primes, before turning to the large sieve and Bombieri's theorem. Before Professor Davenport's untimely death in 1969, several mathematicians had suggested small improvements which might be made in Multiplicative Number Theory, should it ever be reprinted.

• Microsoft Access
• Multiplikative Zahlentheorie
• Prime
• Sieve
• arithmetic
• boundary element method
• number theory
• prime number
• proof
• theorem
• time

From the reviews of the third edition:

"The book under review is one of the most important references in the multiplicative number theory, as its title mentions exactly. … Davenport’s book covers most of the important topics in the theory of distribution of primes and leads the reader to serious research topics … . is very well written. … is useful for graduate students, researchers and for professors. It is a very good text source specially for graduate levels, but even is fruitful for undergraduates." (Mehdi Hassani, MathDL, July, 2008)

• Cambridge University, Cambridge, England

  Harold Davenport

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