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Author: Samuel W. Gilbert
Publisher: Riemann hypothesis
ISBN: 9781439216385
Category : Mathematics
Languages : en
Pages : 160
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Book Description
The author demonstrates that the Dirichlet series representation of the Riemann zeta function converges geometrically at the roots in the critical strip. The Dirichlet series parts of the Riemann zeta function diverge everywhere in the critical strip. It has therefore been assumed for at least 150 years that the Dirichlet series representation of the zeta function is useless for characterization of the non-trivial roots. The author shows that this assumption is completely wrong. Reduced, or simplified, asymptotic expansions for the terms of the zeta function series parts are equated algebraically with reduced asymptotic expansions for the terms of the zeta function series parts with reflected argument, constraining the real parts of the roots of both functions to the critical line. Hence, the Riemann hypothesis is correct. Formulae are derived and solved numerically, yielding highly accurate values of the imaginary parts of the roots of the zeta function.
Author: Samuel W. Gilbert
Publisher: Riemann hypothesis
ISBN: 9781439216385
Category : Mathematics
Languages : en
Pages : 160
Get Book
Book Description
The author demonstrates that the Dirichlet series representation of the Riemann zeta function converges geometrically at the roots in the critical strip. The Dirichlet series parts of the Riemann zeta function diverge everywhere in the critical strip. It has therefore been assumed for at least 150 years that the Dirichlet series representation of the zeta function is useless for characterization of the non-trivial roots. The author shows that this assumption is completely wrong. Reduced, or simplified, asymptotic expansions for the terms of the zeta function series parts are equated algebraically with reduced asymptotic expansions for the terms of the zeta function series parts with reflected argument, constraining the real parts of the roots of both functions to the critical line. Hence, the Riemann hypothesis is correct. Formulae are derived and solved numerically, yielding highly accurate values of the imaginary parts of the roots of the zeta function.
Author: Samuel Gilbert
Publisher:
ISBN: 9781439226872
Category :
Languages : en
Pages : 158
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Book Description
Author: Titchmarch E. C.
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
Author: Harold M. Edwards
Publisher: Courier Corporation
ISBN: 9780486417400
Category : Mathematics
Languages : en
Pages : 338
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Book Description
Superb high-level study of one of the most influential classics in mathematics examines landmark 1859 publication entitled “On the Number of Primes Less Than a Given Magnitude,” and traces developments in theory inspired by it. Topics include Riemann's main formula, the prime number theorem, the Riemann-Siegel formula, large-scale computations, Fourier analysis, and other related topics. English translation of Riemann's original document appears in the Appendix.
Author: Aleksandar Ivic
Publisher: Courier Corporation
ISBN: 0486140040
Category : Mathematics
Languages : en
Pages : 548
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Book Description
This text covers exponential integrals and sums, 4th power moment, zero-free region, mean value estimates over short intervals, higher power moments, omega results, zeros on the critical line, zero-density estimates, and more. 1985 edition.
Author: Peter B. Borwein
Publisher: Springer Science & Business Media
ISBN: 0387721258
Category : Mathematics
Languages : en
Pages : 543
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Book Description
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: Barry Mazur
Publisher: Cambridge University Press
ISBN: 1107101921
Category : Mathematics
Languages : en
Pages : 155
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Book Description
This book introduces prime numbers and explains the famous unsolved Riemann hypothesis.
Author: Naji Arwashan, PhD, PE
Publisher: Nova Science Publishers
ISBN: 1536194220
Category : Mathematics
Languages : en
Pages : 232
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Book Description
This book is an introductory and comprehensive presentation of the Riemann Hypothesis, one of the most important open questions in math today. It is introductory because it is written in an accessible and detailed format that makes it easy to read and understand. And it is comprehensive because it explains and proves all the mathematical ideas surrounding and leading to the formulation of the hypothesis.
Author: Dan Rockmore
Publisher: Vintage
ISBN: 0375727728
Category : Mathematics
Languages : en
Pages : 306
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Book Description
For 150 years the Riemann hypothesis has been the holy grail of mathematics. Now, at a moment when mathematicians are finally moving in on a proof, Dartmouth professor Dan Rockmore tells the riveting history of the hunt for a solution.In 1859 German professor Bernhard Riemann postulated a law capable of describing with an amazing degree of accuracy the occurrence of the prime numbers. Rockmore takes us all the way from Euclid to the mysteries of quantum chaos to show how the Riemann hypothesis lies at the very heart of some of the most cutting-edge research going on today in physics and mathematics.
Author: Michel Laurent Lapidus
Publisher: American Mathematical Soc.
ISBN: 9780821842225
Category : Mathematics
Languages : en
Pages : 594
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Book Description
Formulated in 1859, the Riemann Hypothesis is the most celebrated and multifaceted open problem in mathematics. In essence, it states that the primes are distributed as harmoniously as possible--or, equivalently, that the Riemann zeros are located on a single vertical line, called the critical line.
