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 stimulaยญ 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 ยงยง23-29 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 ยง24). Vinogradov devised his method in order to estimate the sum LPH e(prx); to maintain the historical perspective I have inserted (in ยงยง25, 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
CONTENT
1 Primes in Arithmetic Progression -- 2 Gaussโ{128}{153} Sum -- 3 Cyclotomy -- 4 Primes in Arithmetic Progression: The General Modulus -- 5 Primitive Characters -- 6 Dirichletโ{128}{153}s Class Number Formula -- 7 The Distribution of the Primes -- 8 Riemannโ{128}{153}s Memoir -- 9 The Functional Equation of the L Functions -- 10 Properties of the ? Function -- 11 Integral Functions of Order 1 -- 12 The Infinite Products for ?(s) and ?(s, ?) -- 13 A Zero-Free Region for ?(s) -- 14 Zero-Free Regions for L(s, ?) -- 15 The Number N(T) -- 16 The Number N(T, ?) -- 17 The Explicit Formula for ?(x) -- 18 The Prime Number Theorem -- 19 The Explicit Formula for ?(x, ?) -- 20 The Prime Number Theorem for Arithmetic Progressions (I) -- 21 Siegelโ{128}{153}s Theorem -- 22 The Prime Number Theorem for Arithmetic Progressions (II) -- 23 The Pรณlya-Vinogradov Inequality -- 24 Further Prime Number Sums -- 25 An Exponential Sum Formed with Primes -- 26 Sums of Three Primes -- 27 The Large Sieve -- 28 Bombieriโ{128}{153}s Theorem -- 29 An Average Result -- 30 References to Other Work
