The Theory of the Riemann Zeta-Function (Oxford Science Publications)

The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects of the theory, starting from first principles and probing the function's own challenging theory, with the famous and still unsolved "Riemann hypothesis" at its heart. The second edition has been revised to include descriptions of work done in the last forty years and is updated with many additional references; it will provide stimulating reading for postgraduates and workers in analytic number theory and classical analysis. The Function [zeta](s) And The Dirichlet Series Related To It -- The Analytic Character Of [zeta](s), And The Functional Equations -- The Theorem Of Hadamard And De La Vallée Poussin, And Its Consequences -- Approximate Formulae -- The Order Of [zeta](s) In The Critical Strip -- Vinogradov's Method -- Mean-value Theorems -- [omega]-theorems -- The General Distribution Of Zeros -- The Zeros On The Critical Line -- The General Distribution Of Values Of [zeta](s) -- Divisor Problems -- The Lindelöf Hypothesis -- Consequences Of The Riemann Hypothesis -- Calculations Relating To The Zeros. By E.c. Titchmarsh. Bibliography: P. [392]-412. The Riemann zeta-function is our most important tool in the study of prime numbers, and yet the famous "Riemann hypothesis" at its core remains unsolved. This book studies the theory from every angle and includes new material on recent work.