Eulers Konstante, Primzahlstrände und die Riemannsche Vermutung

Author: Julian Havil

Publisher: Springer-Verlag

ISBN: 3540484965

Category: Mathematics

Page: 302

View: 6340

Jeder kennt p = 3,14159..., viele kennen e = 2,71828..., einige i. Und dann? Die "viertwichtigste" Konstante ist die Eulersche Zahl g = 0,5772156... - benannt nach dem genialen Leonhard Euler (1707-1783). Bis heute ist unbekannt, ob g eine rationale Zahl ist. Das Buch lotet die "obskure" Konstante aus. Die Reise beginnt mit Logarithmen und der harmonischen Reihe. Es folgen Zeta-Funktionen und Eulers wunderbare Identität, Bernoulli-Zahlen, Madelungsche Konstanten, Fettfinger in Wörterbüchern, elende mathematische Würmer und Jeeps in der Wüste. Besser kann man nicht über Mathematik schreiben. Was Julian Havil dazu zu sagen hat, ist spektakulär.

Die Musik der Primzahlen

auf den Spuren des größten Rätsels der Mathematik

Author: Marcus Du Sautoy

Publisher: C.H.Beck

ISBN: 9783406523205

Category: Primzahl

Page: 398

View: 4625


Prime Numbers and the Riemann Hypothesis

Author: Barry Mazur,William Stein

Publisher: Cambridge University Press

ISBN: 1107101921

Category: Mathematics

Page: 150

View: 1237

This book introduces prime numbers and explains the famous unsolved Riemann hypothesis.

Feynmans verschollene Vorlesung

die Bewegung der Planeten um die Sonne

Author: David L. Goodstein,Judith R. Goodstein

Publisher: N.A

ISBN: 9783492229944


Page: 232

View: 2986


The Riemann Hypothesis

A Resource for the Afficionado and Virtuoso Alike

Author: Peter Borwein,Stephen Choi,Brendan Rooney,Andrea Weirathmueller

Publisher: Springer Science & Business Media

ISBN: 0387721258

Category: Mathematics

Page: 533

View: 8005

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: Jan Feliksiak

Publisher: Xlibris Corporation

ISBN: 1479765600

Category: Education

Page: 132

View: 9239

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.

Prime Obsession

Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

Author: John Derbyshire

Publisher: Joseph Henry Press

ISBN: 0309512573

Category: Science

Page: 429

View: 4798

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.


Algebraische Zahlen und Funktionen

Author: Helmut Koch

Publisher: Springer-Verlag

ISBN: 3322803120

Category: Mathematics

Page: 344

View: 7943

Hauptziel des Buches ist die Vermittlung des Grundbestandes der Algebraischen Zahlentheorie einschließlich der Theorie der normalen Erweiterungen bis hin zu einem Ausblick auf die Klassenkörpertheorie. Gleichberechtigt mit algebraischen Zahlen werden auch algebraische Funktionen behandelt. Dies geschieht einerseits um die Analogie zwischen Zahl- und Funktionenkörpern aufzuzeigen, die besonders deutlich im Falle eines endlichen Konstantenkörpers ist. Andererseits erhält man auf diese Weise eine Einführung in die Theorie der "höheren Kongruenzen" als eines wesentlichen Bestandteils der "Arithmetischen Geometrie". Obgleich das Buch hauptsächlich algebraischen Methoden gewidmet ist, findet man in der Einleitung auch einen kurzen Beweis des Primzahlsatzes nach Newman. In den Kapiteln 7 und 8 wird die Theorie der Heckeschen L-Reihen behandelt einschließlich der Verteilung der Primideale algebraischer Zahlkörper in Kegeln.

Casimir Force, Casimir Operators and the Riemann Hypothesis

Mathematics for Innovation in Industry and Science

Author: Gerrit van Dijk,Masato Wakayama

Publisher: Walter de Gruyter

ISBN: 311022612X

Category: Mathematics

Page: 286

View: 3412

This volume contains the proceedings of the conference "Casimir Force, Casimir Operators and the Riemann Hypothesis Mathematics for Innovation in Industry and Science" held in November 2009 in Fukuoka (Japan). The conference focused on the following topics: Casimir operators in harmonic analysis and representation theory Number theory, in particular zeta functions and cryptography Casimir force in physics and its relation with nano-science Mathematical biology Importance of mathematics for innovation in industry "

The Prime Number Theorem

Author: G. J. O. Jameson

Publisher: Cambridge University Press

ISBN: 9780521891103

Category: Mathematics

Page: 252

View: 6104

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.

The Riemann Zeta-Function

Theory and Applications

Author: Aleksandar Ivic

Publisher: Courier Corporation

ISBN: 0486140040

Category: Mathematics

Page: 562

View: 3441

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.

Stalking The Riemann Hypothesis

Author: Daniel Nahum Rockmore

Publisher: Random House

ISBN: 1446483622

Category: Science

Page: 304

View: 3775

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.

Bernhard Riemann 1826–1866

Wendepunkte in der Auffassung der Mathematik

Author: Detlef Laugwitz

Publisher: Springer-Verlag

ISBN: 3034889836

Category: Mathematics

Page: 348

View: 1620

Das Riemannsche Integral lernen schon die Schüler kennen, die Theorien der reellen und der komplexen Funktionen bauen auf wichtigen Begriffsbildungen und Sätzen Riemanns auf, die Riemannsche Geometrie ist für Einsteins Gravitationstheorie und ihre Erweiterungen unentbehrlich, und in der Zahlentheorie ist die berühmte Riemannsche Vermutung noch immer offen. Riemann und sein um fünf Jahre jüngerer Freund Richard Dedekind sahen sich als Schüler von Gauss und Dirichlet. Um die Mitte des 19. Jahrhunderts leiteten sie den Übergang zur "modernen Mathematik" ein, der eine in Analysis und Geometrie, der andere in der Algebra mit der Hinwendung zu Mengen und Strukturen. Dieses Buch ist der erste Versuch, Riemanns wissenschaftliches Werk unter einem einheitlichen Gesichtspunkt zusammenzufassend darzustellen. Riemann gilt als einer der Philosophen unter den Mathematikern. Er stellte das Denken in Begriffen neben die zuvor vorherrschende algorithmische Auffassung von der Mathematik, welche die Gegenstände der Untersuchung, in Formeln und Figuren, in Termumformungen und regelhaften Konstruktionen als die allein legitimen Methoden sah. David Hilbert hat als Riemanns Grundsatz herausgestellt, die Beweise nicht durch Rechnung, sondern lediglich durch Gedanken zu zwingen. Hermann Weyl sah als das Prinzip Riemanns in Mathematik und Physik, "die Welt als das erkenntnistheoretische Motiv..., die Welt aus ihrem Verhalten im un- endlich kleinen zu verstehen."

Prime Numbers

The Most Mysterious Figures in Math

Author: David Wells

Publisher: John Wiley & Sons

ISBN: 0471718920

Category: Mathematics

Page: 288

View: 4022

A fascinating journey into the mind-bending world of prime numbers Cicadas of the genus Magicicada appear once every 7, 13, or 17 years. Is it just a coincidence that these are all prime numbers? How do twin primes differ from cousin primes, and what on earth (or in the mind of a mathematician) could be sexy about prime numbers? What did Albert Wilansky find so fascinating about his brother-in-law's phone number? Mathematicians have been asking questions about prime numbers for more than twenty-five centuries, and every answer seems to generate a new rash of questions. In Prime Numbers: The Most Mysterious Figures in Math, you'll meet the world's most gifted mathematicians, from Pythagoras and Euclid to Fermat, Gauss, and Erd?o?s, and you'll discover a host of unique insights and inventive conjectures that have both enlarged our understanding and deepened the mystique of prime numbers. This comprehensive, A-to-Z guide covers everything you ever wanted to know--and much more that you never suspected--about prime numbers, including: * The unproven Riemann hypothesis and the power of the zeta function * The "Primes is in P" algorithm * The sieve of Eratosthenes of Cyrene * Fermat and Fibonacci numbers * The Great Internet Mersenne Prime Search * And much, much more

Alex im Wunderland der Zahlen

Eine Reise durch die aufregende Welt der Mathematik

Author: Alex Bellos

Publisher: eBook Berlin Verlag

ISBN: 3827078083

Category: Mathematics

Page: 480

View: 1967

Erinnern wir uns nicht alle mit Schrecken an die ratlosen Momente vor der Tafel im Matheunterricht? Mit Kurvendiskussionen und Dreisatz dürften jedenfalls nur wenige Spaß und Spannung verbinden... Bis jetzt! Denn nun wagt sich Alex Bellos in den Kaninchenbau der Mathematik: in das Reich von Geometrie und Algebra, von Wahrscheinlichkeitsrechnung, Statistik und logischen Paradoxa. Auf der anderen Seite des Erdballs, am Amazonas, zählen die Mitglieder des Indianerstammes der Munduruku nur bis fünf und halten die Vorstellung, dass dies nicht genügen solle, für reichlich lächerlich. Bei uns in Deutschland dagegen finden jährlich die Meisterschaften der besten Kopfrechner der Welt statt - 2010 wurde in Magdeburg eine elfjährige Inderin zur Nummer eins unter den "Mathleten" gekürt. Die Mathe-Weltmeisterin unter den Tieren ist hingegen die Schimpansin Ai, die Alex Bellos im japanischen Inuyama aufspürt und über deren Rechenkünste er nur staunen kann. Auch wenn er von den bahnbrechenden Überlegungen Euklids erzählt oder erklärt, warum man in Japan seine Visitenkarten keinesfalls zu Dodekaedern falten sollte - Bellos führt uns durch das wahrhaft erstaunliche Reich der Zahlen und bringt uns eine komplexe Wissenschaft spielerisch nahe. Mit seiner Mischung aus spannender Reportage, Wissenschaftsgeschichte und mathematischen Kabinettstückchen erbringt er souverän den Beweis, dass die Gleichung Mathematik = Langeweile eindeutig nicht wahr ist. Quod erat demonstrandum.

Collected works

Author: Edmund Landau

Publisher: N.A


Category: Algebraic number theory

Page: 415

View: 3587