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In Pursuit of Zeta-3: The World's Most Mysterious Unsolved Math Problem (édition 2021)

par Paul J. Nahin (Auteur)

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An engrossing look at the history and importance of a centuries-old but still unanswered math problem For centuries, mathematicians the world over have tried, and failed, to solve the zeta-3 problem. Math genius Leonhard Euler attempted it in the 1700s and came up short. The straightforward puzzle considers if there exists a simple symbolic formula for the following: 1+(1/2)^3+(1/3)^3+(1/4)^3+. . . . But why is this issue--the sum of the reciprocals of the positive integers cubed--so important? With In Pursuit of Zeta-3, popular math writer Paul Nahin investigates the history and significance of this mathematical conundrum. Drawing on detailed examples, historical anecdotes, and even occasionally poetry, Nahin sheds light on the richness of the nature of zeta-3. He shows its intimate connections to the Riemann hypothesis, another mathematical mystery that has stumped mathematicians for nearly two centuries. He looks at its links with Euler's achievements and explores the modern research area of Euler sums, where zeta-3 occurs frequently. An exact solution to the zeta-3 question wouldn't simply satisfy pure mathematical interest: it would have critical ramifications for applications in physics and engineering, such as quantum electrodynamics. Challenge problems with detailed solutions and MATLAB code are included at the end of each of the book's sections. Detailing the trials and tribulations of mathematicians who have approached one of the field's great unsolved riddles, In Pursuit of Zeta-3 will tantalize curious math enthusiasts everywhere.… (plus d'informations)

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Membre:	Ganzy
Titre:	In Pursuit of Zeta-3: The World's Most Mysterious Unsolved Math Problem
Auteurs:	Paul J. Nahin (Auteur)
Info:	Princeton University Press (2021), 344 pages
Collections:	Votre bibliothèque
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Mots-clés:	mathematics

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In Pursuit of Zeta-3: The World's Most Mysterious Unsolved Math Problem par Paul J. Nahin

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fwalchak, bujeya, vondeuten, Abraxas3d, joopdejong, ray1729, NickEdkins

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Indeholder "Preface", "1. Euler's Problem", " 1.1. Introducing Euler", " 1.2. The Harmonic Series and the Riemann Zeta Function", " 1.3. Euler's Constant, the Zeta Function, and Primes", " 1.4. Euler's Gamma Function, the Reflection Formula, and the Zeta Function", " 1.5. Ramanujan's Master Theorem", " 1.6. Integral Forms for the Harmonic Series and Euler's Constant", " 1.7. Euler's Constant and the Zeta Function Redux (and the Digamma Function, Too)", " 1.8. Calculating ζ(3)", "2. More Wizard Math and the Zeta Function ζ(s)", " 2.1. Euler's Infinite Series for ζ(2)", " 2.2. The Beta Function and the Duplication Formula", " 2.3. Euler Almost Computes ζ(3)", " 2.4. Integral Forms of ζ(2) and ζ(3)", " 2.5. Zeta Near s = 1", " 2.6. Zeta Prime at s = 0", " 2.7. Interlude", "3. Periodic Functions, Fourier Series, and the Zeta Function", " 3.1. The Concept of a Function", " 3.2. Periodic Functions and Their Fourier Series", " 3.3. Complex Fourier Series and Parseval's Power Formula", " 3.4. Calculating ζ(2n) with Fourier Series", " 3.5. How Fourier Series Fail to Compute ζ(3)", " 3.6. Fourier Transforms and Poisson Summation", " 3.7. The Functional Equation of the Zeta Function", "4. Euler Sums, the Harmonic Series, and the Zeta Function", " 4.1. Euler's Original Sums", " 4.2. The Algebra of Euler Sums", " 4.3. Euler's Double Sums", " 4.4. Euler Sums after Euler", "Epilogue", "Appendix 1: Solving the Impossible by Changing the Rules", "Appendix 2: Evaluating Integral_t=0_infinity(e^(-t^2)dt) and Integral_t=0_infinity(e^(-pt^2-q/t^2)dt)", "Appendix 3: Proof That Sum_q=1_infinity ( Sum_n=1_infinity and n ne q (1 /(qn (n-q)))) Equals Zero", "Appendix 4: Double Integration Reversal Isn't Always Legal", "Appendix 5: Impossibility Results from Computer Science", "Challenge Problem Solutions", "Acknowledgments", "Index".

ζ(3) også kaldet zeta(3) eller Apéry's konstant (efter Roger Apéry (1916 - 1994)) er 1.202056903159594285399738161511449990764986292... og er et irrationelt tal, som er summen af reciprokværdierne af alle kubiktallene, dvs 1 + 1/8 + 1/27 + 1/256 ... men man kender ikke nogen eksakt formel for tallet og det er først for nyligt at Apéry har bevist at det er irrationelt. zeta(s) er summen for i = 1 til uendelig af 1/(i ^ s). zeta(2) = pi^2 / 6. zeta af lige heltal er typisk rationelle tal * en potens af pi. Zeta af ulige tal over 1 har man ikke eksakte formler for. Der er en nydelig formel, der beregner zeta(2n) ved hjælp af alle værdierne af zeta(2), zeta(4) osv op til zeta(2n-2).

Her er en række nydelige resultater præsenteret med kommentarer, fx at summen af reciprokværdierne af primtallene også er divergent. Men ganske langsomt. som ln(ln(n)) langsomt.
Fx er summen af reciprokværdierne af den første million primtal kun 3.068219...
Gammafunktionen og Gamma(1/2) = sqrt(pi).
(n!) * (-n!) = n* pi / sin (n*pi) hvilket nåes på lidt lusket vis, men hvis resultatet holder vand, så følger Wallis' produkt nemt. Så det styrker tilliden til resultatet.
Gamma(n)*Gamma(1-n) = pi / sin(n * pi).
Riemann finder en formel for så Gamma(s)*zeta(s) = integralet fra 0 til uendelig af (x^(s-1))/(e^x - 1). Her er også forklaringer på at lave hurtigt konvergerende rækker for fx Eulers konstant. Euler og andre senere matematikere har brugt nogle underholdende cowboy-metoder på zeta(3) problemet. Stort set uden andet end negative resultater. Forfatteren her når langt omkring med omskrivninger af integraler og brug af Fourier-rækker. Men uden at der kommer noget ud af det.
Her er lidt snak (side 172) om det er fourier-rækker, maclaurin-rækker, taylor-rækker eller stirling-rækker. Og om at man jo kan opfatte en vilkårlig ikke-periodisk funktion som en periodisk funktion med en enkelt uendelig lang periode. Riemann-hypotesen bliver også nævnt, men den er svær at foreklare på det elementære niveau, som Nahin sigter efter. Her er fine bemærkninger om Tom Apostol, hvis bog om analytisk talteori, jeg har fulgt et kursus om. Her er også noget Matlab kode strået rundt.
I skrivende stund kende zeta(3) med over 1000 milliarder cifre takket være nogle smarte algoritmer. (

)

bnielsen | Feb 16, 2023 |

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An engrossing look at the history and importance of a centuries-old but still unanswered math problem For centuries, mathematicians the world over have tried, and failed, to solve the zeta-3 problem. Math genius Leonhard Euler attempted it in the 1700s and came up short. The straightforward puzzle considers if there exists a simple symbolic formula for the following: 1+(1/2)^3+(1/3)^3+(1/4)^3+. . . . But why is this issue--the sum of the reciprocals of the positive integers cubed--so important? With In Pursuit of Zeta-3, popular math writer Paul Nahin investigates the history and significance of this mathematical conundrum. Drawing on detailed examples, historical anecdotes, and even occasionally poetry, Nahin sheds light on the richness of the nature of zeta-3. He shows its intimate connections to the Riemann hypothesis, another mathematical mystery that has stumped mathematicians for nearly two centuries. He looks at its links with Euler's achievements and explores the modern research area of Euler sums, where zeta-3 occurs frequently. An exact solution to the zeta-3 question wouldn't simply satisfy pure mathematical interest: it would have critical ramifications for applications in physics and engineering, such as quantum electrodynamics. Challenge problems with detailed solutions and MATLAB code are included at the end of each of the book's sections. Detailing the trials and tribulations of mathematicians who have approached one of the field's great unsolved riddles, In Pursuit of Zeta-3 will tantalize curious math enthusiasts everywhere.

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