Multiplicative Number Theory

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 {sect}{sect}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 {sect}24). Vinogradov devised his method in order to estimate the sum LPH e(prx); to maintain the historical perspective I have inserted (in {sect}{sect}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 Front Matter....Pages i-xiii Primes in Arithmetic Progression....Pages 1-11 Gauss’ Sum....Pages 12-16 Cyclotomy....Pages 17-26 Primes in Arithmetic Progression: The General Modulus....Pages 27-34 Primitive Characters....Pages 35-42 Dirichlet’s Class Number Formula....Pages 43-53 The Distribution of the Primes....Pages 54-58 Riemann’s Memoir....Pages 59-64 The Functional Equation of the L Functions....Pages 65-72 Properties of the Γ Function....Pages 73-73 Integral Functions of Order 1....Pages 74-78 The Infinite Products for ξ( s ) and ξ( s, χ )....Pages 79-83 A Zero-Free Region for ζ(s)....Pages 84-87 Zero-Free Regions for L ( s, χ )....Pages 88-96 The Number N ( T )....Pages 97-100 The Number N ( T, χ )....Pages 101-103 The Explicit Formula for Ψ( x )....Pages 104-110 The Prime Number Theorem....Pages 111-114 The Explicit Formula for ψ( x , χ)....Pages 115-120 The Prime Number Theorem for Arithmetic Progressions (I)....Pages 121-125 Siegel’s Theorem....Pages 126-131 The Prime Number Theorem for Arithmetic Progressions (II)....Pages 132-134 The Pólya-Vinogradov Inequality....Pages 135-137 Further Prime Number Sums....Pages 138-142 An Exponential Sum Formed with Primes....Pages 143-144 Sums of Three Primes....Pages 145-150 The Large Sieve....Pages 151-160 Bombieri’s Theorem....Pages 161-168 An Average Result....Pages 169-171 References to other Work....Pages 172-174 Back Matter....Pages 175-177 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 stimulation, 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 chapters 23-29 completely for the second edition. In making these alterations I have tried to preserve the tone and spirit of the original. Before Professor Davenport's untimely death in 1969, several mathematicians had suggested small improvements which might be made in the text, should it ever be reprinted. Most of these have been incorporated here. Finally, the mathematical community is indebted to Professor J.-P. Serre for urging Springer-Verlag to publish a new edition of this important book. -- H.L.M