The Riemann Hypothesis and Hilbert's Tenth Problem
Author | : Sarvadaman Chowla |
Publisher | : CRC Press |
Total Pages | : 144 |
Release | : 1987 |
Genre | : Mathematics |
ISBN | : 9780677001401 |
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Author | : Sarvadaman Chowla |
Publisher | : CRC Press |
Total Pages | : 144 |
Release | : 1987 |
Genre | : Mathematics |
ISBN | : 9780677001401 |
Author | : Peter B. Borwein |
Publisher | : Springer Science & Business Media |
Total Pages | : 543 |
Release | : 2008 |
Genre | : Mathematics |
ISBN | : 0387721258 |
The Riemann Hypothesis has become the Holy Grail of mathematics in the century and a half since 1859 when Bernhard Riemann, one of the extraordinary mathematical talents of the 19th century, originally posed the problem. While the problem is notoriously difficult, and complicated even to state carefully, it can be loosely formulated as "the number of integers with an even number of prime factors is the same as the number of integers with an odd number of prime factors." The Hypothesis makes a very precise connection between two seemingly unrelated mathematical objects, namely prime numbers and the zeros of analytic functions. If solved, it would give us profound insight into number theory and, in particular, the nature of prime numbers. This book is an introduction to the theory surrounding the Riemann Hypothesis. Part I serves as a compendium of known results and as a primer for the material presented in the 20 original papers contained in Part II. The original papers place the material into historical context and illustrate the motivations for research on and around the Riemann Hypothesis. Several of these papers focus on computation of the zeta function, while others give proofs of the Prime Number Theorem, since the Prime Number Theorem is so closely connected to the Riemann Hypothesis. The text is suitable for a graduate course or seminar or simply as a reference for anyone interested in this extraordinary conjecture.
Author | : M. Ram Murty |
Publisher | : American Mathematical Soc. |
Total Pages | : 256 |
Release | : 2019-05-09 |
Genre | : Mathematics |
ISBN | : 1470443996 |
Hilbert's tenth problem is one of 23 problems proposed by David Hilbert in 1900 at the International Congress of Mathematicians in Paris. These problems gave focus for the exponential development of mathematical thought over the following century. The tenth problem asked for a general algorithm to determine if a given Diophantine equation has a solution in integers. It was finally resolved in a series of papers written by Julia Robinson, Martin Davis, Hilary Putnam, and finally Yuri Matiyasevich in 1970. They showed that no such algorithm exists. This book is an exposition of this remarkable achievement. Often, the solution to a famous problem involves formidable background. Surprisingly, the solution of Hilbert's tenth problem does not. What is needed is only some elementary number theory and rudimentary logic. In this book, the authors present the complete proof along with the romantic history that goes with it. Along the way, the reader is introduced to Cantor's transfinite numbers, axiomatic set theory, Turing machines, and Gödel's incompleteness theorems. Copious exercises are included at the end of each chapter to guide the student gently on this ascent. For the advanced student, the final chapter highlights recent developments and suggests future directions. The book is suitable for undergraduates and graduate students. It is essentially self-contained.
Author | : Jan Denef |
Publisher | : American Mathematical Soc. |
Total Pages | : 384 |
Release | : 2000 |
Genre | : Mathematics |
ISBN | : 0821826220 |
This book is the result of a meeting that took place at the University of Ghent (Belgium) on the relations between Hilbert's tenth problem, arithmetic, and algebraic geometry. Included are written articles detailing the lectures that were given as well as contributed papers on current topics of interest. The following areas are addressed: an historical overview of Hilbert's tenth problem, Hilbert's tenth problem for various rings and fields, model theory and local-global principles, including relations between model theory and algebraic groups and analytic geometry, conjectures in arithmetic geometry and the structure of diophantine sets, for example with Mazur's conjecture, Lang's conjecture, and Bücchi's problem, and results on the complexity of diophantine geometry, highlighting the relation to the theory of computation. The volume allows the reader to learn and compare different approaches (arithmetical, geometrical, topological, model-theoretical, and computational) to the general structural analysis of the set of solutions of polynomial equations. It would make a nice contribution to graduate and advanced graduate courses on logic, algebraic geometry, and number theory
Author | : I︠U︡riĭ V. Matii︠a︡sevich |
Publisher | : MIT Press |
Total Pages | : 296 |
Release | : 1993 |
Genre | : Computers |
ISBN | : 9780262132954 |
This book presents the full, self-contained negative solution of Hilbert's 10th problem.
Author | : Barry Mazur |
Publisher | : Cambridge University Press |
Total Pages | : 155 |
Release | : 2016-04-11 |
Genre | : Mathematics |
ISBN | : 1107101921 |
This book introduces prime numbers and explains the famous unsolved Riemann hypothesis.
Author | : Karl Sabbagh |
Publisher | : Macmillan |
Total Pages | : 364 |
Release | : 2003 |
Genre | : Mathematics |
ISBN | : 9780374250072 |
An engaging, informative, and wryly humorous exploration of one of the great conundrums of all time In 1859 Bernhard Riemann, a shy German mathematician, wrote an eight-page article giving an answer to a problem that had long puzzled mathematicians. But he didn’t provide a proof. In fact, he said he couldn’t prove it but he thought that his answer was “very probably” true. From the publication of that paper to the present day, the world’s mathematicians have been fascinated, infuriated, and obsessed with proving the Riemann Hypothesis, and so great is the interest in its solution that in 2001 an American foundation put up prize money of $1 million for the first person to demonstrate that the hypothesis is correct. The hypothesis refers to prime numbers, which are in some sense the atoms from which all other numbers are constructed, and seeks to explain where every single prime to infinity will occur. Riemann’s idea—if true—would illuminate how these numbers are distributed, and if false will throw pure mathematics into confusion. Karl Sabbagh meets some of the world’s mathematicians who spend their lives thinking about the Riemann Hypothesis, focusing attention in particular on “Riemann’s zeros,” a series of points that are believed to lie in a straight line, though no one can prove it. Accessible and vivid, The Riemann Hypothesis is a brilliant explanation of numbers and a profound meditation on the ultimate meaning of mathematics.
Author | : Eugenio G. Omodeo |
Publisher | : Springer |
Total Pages | : 454 |
Release | : 2017-01-27 |
Genre | : Philosophy |
ISBN | : 3319418424 |
This book presents a set of historical recollections on the work of Martin Davis and his role in advancing our understanding of the connections between logic, computing, and unsolvability. The individual contributions touch on most of the core aspects of Davis’ work and set it in a contemporary context. They analyse, discuss and develop many of the ideas and concepts that Davis put forward, including such issues as contemporary satisfiability solvers, essential unification, quantum computing and generalisations of Hilbert’s tenth problem. The book starts out with a scientific autobiography by Davis, and ends with his responses to comments included in the contributions. In addition, it includes two previously unpublished original historical papers in which Davis and Putnam investigate the decidable and the undecidable side of Logic, as well as a full bibliography of Davis’ work. As a whole, this book shows how Davis’ scientific work lies at the intersection of computability, theoretical computer science, foundations of mathematics, and philosophy, and draws its unifying vision from his deep involvement in Logic.
Author | : Kevin Broughan |
Publisher | : Cambridge University Press |
Total Pages | : 705 |
Release | : 2023-09-30 |
Genre | : Mathematics |
ISBN | : 1009384805 |
This third volume presents further equivalents to the Riemann hypothesis and explores its decidability.
Author | : Kevin Broughan |
Publisher | : Cambridge University Press |
Total Pages | : 706 |
Release | : 2023-09-30 |
Genre | : Mathematics |
ISBN | : 1009384775 |
This three-volume work presents the main known equivalents to the Riemann hypothesis, perhaps the most important problem in mathematics. Volume 3 covers new arithmetic and analytic equivalences from numerous studies in the field, such as Rogers and Tao, and presents derivations which show whether the Riemann hypothesis is decidable.