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Mathematicians have recently made dramatic progress on the Twin Primes Conjecture, which asserts that there are infinitely many pairs of prime numbers that differ by 2. This book will describe two stories: that of the recent work on the Twin Primes Conjecture, and in parallel the related ideas from the previous two thousand years of mathematics.--

Numbers, Prime.

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The Riemann hypothesis (RH) is perhaps the most important outstanding problem in mathematics. This two-volume text presents the main known equivalents to RH using analytic and computational methods. The book is gentle on the reader with definitions repeated, proofs split into logical sections, and graphical descriptions of the relations between different results. It also includes extensive tables, supplementary computational tools, and open problems suitable for research. Accompanying software is free to download. These books will interest mathematicians who wish to update their knowledge, graduate and senior undergraduate students seeking accessible research problems in number theory, and others who want to explore and extend results computationally. Each volume can be read independently. Volume 1 presents classical and modern arithmetic equivalents to RH, with some analytic methods. Volume 2 covers equivalences with a strong analytic orientation, supported by an extensive set of appendices containing fully developed proofs.

Riemann hypothesis. --- Numbers, Prime. --- Number theory. --- Riemann, Bernhard,

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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