The Distribution of Prime NumbersOriginally published in 1934 in the Cambridge Tracts this volume presents the theory of the distribution of the prime numbers in the series of natural numbers. The major part of the book is devoted to the analytical theory founded on the zeta-function of Riemann. Despite being long out of print, this Tract still remains unsurpassed as an introduction to the field, combining an economy of detail with a clarity of exposition which eases the novice into this area. |
Contents
Introduction | 1 |
Elementary Theorems | 9 |
The Prime Number Theorem | 25 |
Further Theory of 𝛇 ₈ Applications | 41 |
Explicit Formulae | 68 |
Irregularities of Distribution | 86 |
| 108 | |
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Common terms and phrases
A₁ A₂ absolutely convergent analytic continuation analytic function applications C₁ Cauchy's theorem Chapter Chebyshev complex variable consecutive primes constant deduce defined denote difference between consecutive Dirichlet's series distribution of primes elementary equivalent Euler's identity exists explicit formula factor fixed positive number follows from Theorem functional equation G. H. Hardy half-plane Hardy and Littlewood Hence inequality Ingham integral function integrand Landau Lemma log log log log² log²x logarithmic logx Math maximum modulus principle method number of primes o+ti obtain points positive integers prime number theorem proof properties prove real axis relation Riemann hypothesis s-plane Selberg sieve theory simple pole sufficiently large summation suppose t₁ t₂ Tauberian Theorem 19 Theorem 25 Theorem 34 Theorem H Titchmarsh tract uniformly convergent Vallée Poussin Weierstrass's x₁ zeta-function