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A Computer-Algebra-Based Formal Proof of the Irrationality of ζ(3) [chapter]

Frédéric Chyzak, Assia Mahboubi, Thomas Sibut-Pinote, Enrico Tassi
2014 Lecture Notes in Computer Science  
This paper describes the formal verification of an irrationality proof of ζ(3), the evaluation of the Riemann zeta function, using the Coq proof assistant.  ...  We formally prove this result by an a posteriori verification of calculations performed by computer algebra algorithms in a Maple session.  ...  Formal proof of the common recurrence This section describes the computer-algebra-aided formal proof of Lemma 2, based on a Maple session implementing the program described in Table 1 .  ... 
doi:10.1007/978-3-319-08970-6_11 fatcat:mlorqqxcxbb7dpjx5i53dfmudu

A Formal Proof of the Irrationality of ζ(3) [article]

Assia Mahboubi, Thomas Sibut-Pinote
2021 arXiv   pre-print
This paper presents a complete formal verification of a proof that the evaluation of the Riemann zeta function at 3 is irrational, using the Coq proof assistant.  ...  We formally prove this result by an a posteriori verification of calculations performed by computer algebra algorithms in a Maple session.  ...  We also thank Cyril Cohen, Pierre Roux and Enrico Tassi for their help, in particular with the libraries this work depends on.  ... 
arXiv:1912.06611v6 fatcat:kijni3uvvjcpjhvm4esnq3bcgy

I Prefer Pi: A Brief History and Anthology of Articles in the American Mathematical Monthly

Jonathan M. Borwein and Scott T. Chapman, Scott T. Chapman
2015 The American mathematical monthly  
In celebration of both a special "big" π Day (3/14/15) and the 2015 centennial of the Mathematical Association of America, we review the illustrious history of the constant π in the pages of the American  ...  The authors thank Ivars Peterson for his help in assembling our bibliography.  ...  Last but not least, they thank two former MONTHLY editors, Roger Horn and Dan Velleman, for extensive comments on an earlier draft of this paper.  ... 
doi:10.4169/amer.math.monthly.122.03.195 fatcat:bbjeapvynvhezch46twpkwyi24

Computerized deconstruction

Doron Zeilberger
2003 Advances in Applied Mathematics  
The inequality (DERRIDA + TURING) > (DERRIDA) + (TURING) will be illustrated by computerized deconstruction of Roger Apéry's miraculous proofs of irrationality.  ...  non-trivial steps in Apéry's proofs of the irrationality proofs of log 2, ζ(2) and ζ(3), respectively.  ...  I will illustrate this methodology by deconstructing, in four different ways, one of my all-time favorites: Roger Apéry's proof of the irrationality of ζ (3) .  ... 
doi:10.1016/s0196-8858(02)00508-0 fatcat:3357mbq5avavljilhkees4547m

WHAT IS... a Period?

Stefan Müller-Stach
2014 Notices of the American Mathematical Society  
Even for the odd zeta-values ζ(3), ζ(5), . . . only a few results about their irrationality are known. By work of Apéry, ζ(3) is irrational.  ...  The pro-algebraic torsor Isom ⊗ (T dR , T sing ) is given by Spec( P formal ), where P formal is the algebra of formal periods ( i.e., generated by quadruples (X, D, ω, γ)), and subject only to the relations  ... 
doi:10.1090/noti1159 fatcat:sex5apmc3zfcladgqnhp5he3xa

Infinitesimal Thurston Rigidity and the Fatou-Shishikura Inequality [article]

Adam Epstein
1999 arXiv   pre-print
We prove a refinement of the Fatou-Shishikura Inequality - that the total count of nonrepelling cycles of a rational map is less than or equal to the number of independent infinite forward critical orbits  ...  - from a suitable application of Thurston's Rigidity Theorem - the injectivity of I-f_* on spaces of meromorphic quadratic differentials.  ...  The space D x (f ) of invariant divergences is computed in terms of the formal invariants of x : • If x is superattracting then D x (f ) = 0. • If x is attracting, repelling or irrationally indifferent  ... 
arXiv:math/9902158v1 fatcat:qerdcgnkd5c3lf6r5xlu52y5yq

I prefer pi: A brief history and anthology of articles in the American Mathematical Monthly (2015) [chapter]

Jonathan M. Borwein
2016 Pi: The Next Generation  
In celebration of both a special "big" π Day (3/14/15) and the 2015 centennial of the Mathematical Association of America, we review the illustrious history of the constant π in the pages of the American  ...  The authors thank Ivars Peterson for his help in assembling our bibliography.  ...  Last but not least, they thank two former MONTHLY editors, Roger Horn and Dan Velleman, for extensive comments on an earlier draft of this paper.  ... 
doi:10.1007/978-3-319-32377-0_25 fatcat:w3za7435hnfgnaky4pekrfrr3y

Principal solutions of recurrence relations and irrationality questions in number theory [article]

Angelo B. Mingarelli
2008 arXiv   pre-print
new series expansions of powers of ζ(3) and ζ(2) in terms of Apéry's now classic sequences.  ...  When applied to the situation of powers of ζ(3) it is not known whether the corresponding four term recurrence relation does or does not have such a principal solution, however the method does generate  ...  The simplest consequences involve yet another interpretation of the proof of the irrationality of ζ(3) (and of ζ (2) ).  ... 
arXiv:math/0608577v2 fatcat:twl7qvhtbrgwzpsl4gkoyhepzy

Apéry Limits: Experiments and Proofs [article]

Marc Chamberland, Armin Straub
2020 arXiv   pre-print
An important component of Apéry's proof that ζ (3) is irrational involves representing ζ (3) as the limit of the quotient of two rational solutions to a three-term recurrence.  ...  In the spirit of Jon Borwein, we advertise an experimental-mathematics approach by first exploring in detail a simple but instructive motivating example. We conclude with various open problems.  ...  We are grateful to Alan Sokal for improving the exposition by kindly sharing lots of careful suggestions and comments.  ... 
arXiv:2011.03400v1 fatcat:5jvups4lwzdzrchoxb5e7yh2um

Arithmetic hypergeometric series

Wadim V Zudilin
2011 Russian Mathematical Surveys  
Originally designed as a tool for solving these problems, the hypergeometric series have become a connecting link between different areas of number theory and mathematics in general.  ...  The main goal of our survey is to give common characteristics of auxiliary hypergeometric functions (and their generalisations), functions which occur in number-theoretical problems.  ...  Apéry's sequence used in his proof [12] of the irrationality of ζ(3).  ... 
doi:10.1070/rm2011v066n02abeh004742 fatcat:o6c6ny5q5nbalpj6qreujrecdq

The Life of π: From Archimedes to ENIAC and Beyond [chapter]

Jonathan M. Borwein
2013 From Alexandria, Through Baghdad  
The desire to understand π, the challenge, and originally the need, to calculate ever more accurate values of π, the ratio of the circumference of a circle to its diameter, has challenged mathematicians-great  ...  and less great-for many many centuries and, especially recently, π has provided compelling examples of computational mathematics.  ...  We next give, in extenso, Ivan Niven's 1947 short proof of the irrationality of π.  ... 
doi:10.1007/978-3-642-36736-6_24 fatcat:znmfjjgqrze6bkjhdxbx52db5u

Rational approximations to algebraic Laurent series with coefficients in a finite field

Alina Firicel
2013 Acta Arithmetica  
As an illustration, we give few examples of algebraic Laurent series for which we are able to compute the exact value of the irrationality exponent.  ...  In this paper we give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field.  ...  The aim of this paper is to introduce a new approach in order to bound up the irrationality exponent of an algebraic Laurent series, which is based on the use of the Laurent series expansion.  ... 
doi:10.4064/aa157-4-1 fatcat:ruouxaxoxzf7ljmgubzwjhondm

Rational approximations to algebraic Laurent series with coefficients in a finite field [article]

Alina Firicel
2011 arXiv   pre-print
As an illustration, we give few examples of algebraic Laurent series for which we are able to compute the exact value of the irrationality exponent.  ...  In this paper we give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field.  ...  The aim of this paper is to introduce a new approach in order to bound up the irrationality exponent of an algebraic Laurent series, which is based on the use of the Laurent series expansion.  ... 
arXiv:1102.5764v1 fatcat:niou6ac5azgkzjrgbysoe24h7m

Apéry constants of homogeneous varieties [article]

Sergey Galkin
2016 arXiv   pre-print
We do numerical computations in case of homogeneous varieties. These numbers are identified to be polynomials in the values of Riemann zeta-function with natural arguments.  ...  For Fano manifolds we define Ap\'ery constants and Ap\'ery class as particular limits of ratios of coefficients of solutions of the quantum differential equation.  ...  Golyshev's deresonance computation [7] of the first Apéry constant for Gr(2, N).  ... 
arXiv:1604.04652v1 fatcat:knumot5lcnckticghjxgyddu7e

Pseudo-unitary non-self-dual fusion categories of rank 4 [article]

Hannah K. Larson
2014 arXiv   pre-print
In doing so, we classify all based rings associated with near-group categories of the group Z/3Z.  ...  More precisely, we show that if C is such a fusion category, then its Grothendieck ring K(C) must be one of seven based rings, six of which have know categorifications.  ...  I would also like to thank my brother, Eric Larson, for teaching me about the theory of cyclotomic fields (such as some of the methods used in section 6) and proof-reading the paper.  ... 
arXiv:1401.1879v2 fatcat:vkylxbtfrvaibhbukeejhvgaba
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