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This book has grown out of lectures given at Oxford in 1970 and at University College, Cardiff, intended in each ease for graduate students as an introduction to analytic number theory. The lectures were based on Davenport’s Multiplicative Number Theory, but incorporated simplifications in several proofs, recent work, and other extra material.
Analytic number theory, whilst containing a diversity of results, has one unifying method: that of uniform distribution, mediated by certain sums, which may be exponential sums, character sums, or Dirichlet polynomials, according to the type of uniform distribution required. The study of prime numbers leads to all three. Hopes of elegant asymptotic formulae are dashed by the existence of complex zeros of the Riemann zeta function and of the Dirichlet L-functions. The prime-number theorem depends on the qualitative result that all zeros have real parts less than one. A zero-density theorem is a quantitative result asserting that not many zeros have real parts close to one. In recent years many problems concerning prime numbers have been reduced to that of obtaining a sufficiently strong zero-density theorem.
The first part of this book is introductory in nature it presents the notions of uniform distribution and of large sieve inequalities. In the second part the theory of the zeta function and L-functions is developed and the prime-number theorem proved. The third part deals with large sieve results and mean-value theorems for L-functions, and these are used in the fourth part to prove the main results. These are the theorem of Bombieri and A.I. Vinogradov on primes in arithmetic progressions, a result on gaps between prime numbers, and I.M. Vinogradov’s theorem that every large odd number is a sum of three primes. The treatment is self-contained as far as possible a few results are quoted from Hardy and Wright (1960) and from Titchmarsh (1951).
Parts of prime-number theory not touched here, such as the problem of the least prime in an arithmetical progression, are treated in Prachar’s Primzahlverteilung (Springer 1957). Further work on zero-density theorems is to be found in Montgomery (1971), who also gives a wide list of references covering the field