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Important results surrounding the proof of Goldbach's ternary conjecture are presented in this book. Beginning with an historical perspective along with an overview of essential lemmas and theorems, this monograph moves on to a detailed proof of Vinogradov's theorem. The principles of the Hardy-Littlewood circle method are outlined and applied to Goldbach's ternary conjecture. New results due to H. Maier and the author on Vinogradov's theorem are proved under the assumption of the Riemann hypothesis. The final chapter discusses an approach to Goldbach's conjecture through theorems by L. G. Schnirelmann. This book concludes with an Appendix featuring a sketch of H. Helfgott's proof of Goldbach's ternary conjecture. The Appendix also presents some biographical remarks of mathematicians whose research has played a seminal role on the Goldbach ternary problem. The author's step-by-step approach makes this book accessible to those that have mastered classical number theory and fundamental notions of mathematical analysis. This book will be particularly useful to graduate students and mathematicians in analytic number theory, approximation theory as well as to researchers working on Goldbach's problem.

Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Numerical analysis. --- Number theory. --- Number Theory. --- Analysis. --- Numerical Analysis. --- Number study --- Numbers, Theory of --- Algebra --- Mathematical analysis --- 517.1 Mathematical analysis --- Math --- Science --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic

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This book is concerned with the Riemann Zeta Function, its generalizations, and various applications to several scientific disciplines, including Analytic Number Theory, Harmonic Analysis, Complex Analysis and Probability Theory. Eminent experts in the field illustrate both old and new results towards the solution of long-standing problems and include key historical remarks. Offering a unified, self-contained treatment of broad and deep areas of research, this book will be an excellent tool for researchers and graduate students working in Mathematics, Mathematical Physics, Engineering and Cryptography.

Mathematics. --- Algebraic geometry. --- Harmonic analysis. --- Difference equations. --- Functional equations. --- Dynamics. --- Ergodic theory. --- Functions of complex variables. --- Number theory. --- Number Theory. --- Algebraic Geometry. --- Functions of a Complex Variable. --- Dynamical Systems and Ergodic Theory. --- Difference and Functional Equations. --- Abstract Harmonic Analysis. --- Complex variables --- Ergodic transformations --- Dynamical systems --- Kinetics --- Equations, Functional --- Calculus of differences --- Differences, Calculus of --- Equations, Difference --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Algebraic geometry --- Math --- Number study --- Numbers, Theory of --- Riemann hypothesis. --- Riemann's hypothesis --- Numbers, Prime --- Geometry, algebraic. --- Differentiable dynamical systems. --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Functional analysis --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Elliptic functions --- Functions of real variables --- Geometry --- Algebra --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics