The distribution of prime numbers

Abstract

CONTENTS Introduction Chapter I. The Riemann zeta-function and its connection with primes § 1. Definition of and Euler's identity § 2. Continuation of to the half-plane § 3. Continuation of to the whole plane § 4. Functional properties of and § 5. Zeros of and primes § 6. Elementary theorems on the complex zeros of § 7. Theorems of de la Vallée Poussin § 8. Elementary consequences of the Riemann hypothesis Notes Chapter II. Approximate functional equations for § 1. An approximation of an exponential sum by a sum of integrals § 2. An asymptotic calculation for a class of exponential integrals § 3. An approximation of an exponential sum by a shorter one § 4. Approximate functional equations for Notes Chapter III. Vinogradov's method in the theory of the Riemann zeta-function § 1. The mean-value of a power of the modulus of an exponential sum § 2. Simple lemmas § 3. The main recurrence inequality § 4. Vinogradov's mean-value theorem § 5. An estimate for a zeta-sum and its consequences § 6. The current boundary of the zero-free region and some consequences Notes Chapter IV. A zero-density theorem and primes in short intervals § 1. Auxiliary lemmas § 2. A zero-density theorem § 3. Primes in short intervals Notes Appendix References