The Distribution of Prime Numbers

½π A₁ A₂ log absolutely convergent analytic continuation analytic function applications C₁ C₂ Cauchy's theorem chapter Chebyshev complex variable constant deduce defined denote Dirichlet's series distribution of primes elementary 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 infer integral function integrand Landau Lemma log² log²x logarithmic logx maximum modulus principle method notation number of primes o+ti obtain points positive integers prime number theorem proof prove real axis relation Riemann hypothesis s-plane simple pole sufficiently large summation suppose t₁ t₂ Tauberian tends Theorem 19 Theorem 25 Theorem 34 Theorem H theory uniformly convergent Vallée Poussin values Weierstrass's x₁ xs+1 zeta-function Σπί