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A Computer-Algebra-Based Formal Proof of the Irrationality of ζ(3)
[chapter]
2014
Lecture Notes in Computer Science
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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]
2021
arXiv
pre-print
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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
2015
The American mathematical monthly
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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
2003
Advances in Applied Mathematics
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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?
2014
Notices of the American Mathematical Society
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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]
1999
arXiv
pre-print
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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]
2016
Pi: The Next Generation
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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]
2008
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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
2011
Russian Mathematical Surveys
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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]
2013
From Alexandria, Through Baghdad
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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
2013
Acta Arithmetica
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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]
2011
arXiv
pre-print
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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]
2016
arXiv
pre-print
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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]
2014
arXiv
pre-print
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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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