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Author : Peter Borwein
ISBN : 9780387721255
Genre : Mathematics
File Size : 87.22 MB
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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 : G. J. O. Jameson
ISBN : 0521891108
Genre : Mathematics
File Size : 70.85 MB
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The prime numbers appear to be distributed in a very irregular way amongst the integers, but the prime number theorem provides a simple formula that tells us (in an approximate but well-defined sense) how many primes we can expect to find that are less than any integer we might choose. This is indisputably one of the the great classical theorems of mathematics. Suitable for advanced undergraduates and beginning graduates, this textbook demonstrates how the tools of analysis can be used in number theory to attack a famous problem.

Author : Richard Crandall
ISBN : 9780387289793
Genre : Mathematics
File Size : 82.77 MB
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Bridges the gap between theoretical and computational aspects of prime numbers Exercise sections are a goldmine of interesting examples, pointers to the literature and potential research projects Authors are well-known and highly-regarded in the field

Author : Jan Feliksiak
ISBN : 9781479765607
Genre : Education
File Size : 22.67 MB
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This book presents research results concerning the distribution of prime numbers. The first major result discussed is the supremum for the maximal prime gaps. By an implementation of a binomial coefficient the maximal prime gaps supremum bound is proved, simultaneously establishing the infimum for primes in the short interval. Subsequently, a novel application of the theory of the primorial function establishes the tailored logarithmic integral, which is a superior adaptation of the classical Gauss' logarithmic integral. The tailored integral due to its radically improved accuracy over the Gauss' logarithmic integral, constitutes the supremum bound of estimation of the prime counting function. It presents the possibility to estimate the prime counting function with unprecedented accuracy.

Author : John Derbyshire
ISBN : 9780309512572
Genre : Science
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In August 1859 Bernhard Riemann, a little-known 32-year old mathematician, presented a paper to the Berlin Academy titled: "On the Number of Prime Numbers Less Than a Given Quantity." In the middle of that paper, Riemann made an incidental remark a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years. Today, after 150 years of careful research and exhaustive study, the question remains. Is the hypothesis true or false? Riemann's basic inquiry, the primary topic of his paper, concerned a straightforward but nevertheless important matter of arithmetic defining a precise formula to track and identify the occurrence of prime numbers. But it is that incidental remark the Riemann Hypothesis that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant shrouded in the shadows subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age. It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp, holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. Hunting down the solution to the Riemann Hypothesis has become an obsession for many the veritable "great white whale" of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution. Alternating passages of extraordinarily lucid mathematical exposition with chapters of elegantly composed biography and history, Prime Obsession is a fascinating and fluent account of an epic mathematical mystery that continues to challenge and excite the world. Posited a century and a half ago, the Riemann Hypothesis is an intellectual feast for the cognoscenti and the curious alike. Not just a story of numbers and calculations, Prime Obsession is the engrossing tale of a relentless hunt for an elusive proof and those who have been consumed by it.

Author : Anatoly A. Karatsuba
ISBN : 9783110886146
Genre : Mathematics
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The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany

Author : Roland van der Veen
ISBN : 9780883856505
Genre : Mathematics
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This book introduces interested readers to one of the most famous and difficult open problems in mathematics: the Riemann Hypothesis. Finding a proof will not only make you famous, but also earns you a one million dollar prize. The book originated from an online internet course at the University of Amsterdam for mathematically talented secondary school students. Its aim was to bring them into contact with challenging university level mathematics and show them why the Riemann Hypothesis is such an important problem in mathematics. After taking this course, many participants decided to study in mathematics at university.