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Author : Daniel Nahum Rockmore
ISBN : 9781446483626
Genre : Science
File Size : 43.15 MB
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Like a hunter who sees 'a bit of blood' on the trail, that's how Princeton mathematician Peter Sarnak describes the feeling of chasing an idea that seems to have a chance of success. If this is so, then the jungle of abstractions that is mathematics is full of frenzied hunters these days. They are out stalking big game: the resolution of 'The Riemann Hypothesis', seems to be in their sights. The Riemann Hypothesis is about the prime numbers, the fundamental numerical elements. Stated in 1859 by Professor Bernhard Riemann, it proposes a simple law which Riemann believed a 'very likely' explanation for the way in which the primes are distributed among the whole numbers, indivisible stars scattered without end throughout a boundless numerical universe. Just eight years later, at the tender age of thirty-nine Riemann would be dead from tuberculosis, cheated of the opportunity to settle his conjecture. For over a century, the Riemann Hypothesis has stumped the greatest of mathematical minds, but these days frustration has begun to give way to excitement. This unassuming comment is revealing astounding connections among nuclear physics, chaos and number theory, creating a frenzy of intellectual excitement amplified by the recent promise of a one million dollar bounty. The story of the quest to settle the Riemann Hypothesis is one of scientific exploration. It is peopled with solitary hermits and gregarious cheerleaders, cool calculators and wild-eyed visionaries, Nobel Prize-winners and Fields Medalists. To delve into the Riemann Hypothesis is to gain a window into the world of modern mathematics and the nature of mathematics research. Stalking the Riemann Hypothesis will open wide this window so that all may gaze through it in amazement.

Author : Jan Feliksiak
ISBN : 9781479765607
Genre : Education
File Size : 59.54 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 : Peter Borwein
ISBN : 9780387721255
Genre : Mathematics
File Size : 53.63 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 : 73.24 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 : S. J. Patterson
ISBN : 0521499054
Genre : Mathematics
File Size : 66.65 MB
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This is a modern introduction to the analytic techniques used in the investigation of zeta-function. Riemann introduced this function in connection with his study of prime numbers, and from this has developed the subject of analytic number theory. Since then, many other classes of "zeta-function" have been introduced and they are now some of the most intensively studied objects in number theory. Professor Patterson has emphasized central ideas of broad application, avoiding technical results and the customary function-theoretic approach.