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

2014
*
Lecture Notes in Computer Science
*

This paper describes

doi:10.1007/978-3-319-08970-6_11
fatcat:mlorqqxcxbb7dpjx5i53dfmudu
*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 . ...##
###
A Formal Proof of the Irrationality of ζ(3)
[article]

2021
*
arXiv
*
pre-print

This paper presents

arXiv:1912.06611v6
fatcat:kijni3uvvjcpjhvm4esnq3bcgy
*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. ...##
###
I Prefer Pi: A Brief History and Anthology of Articles in the American Mathematical Monthly

2015
*
The American mathematical monthly
*

In celebration

doi:10.4169/amer.math.monthly.122.03.195
fatcat:bbjeapvynvhezch46twpkwyi24
*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. ...##
###
Computerized deconstruction

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*) . ...

##
###
WHAT IS... a Period?

2014
*
Notices of the American Mathematical Society
*

Even for

doi:10.1090/noti1159
fatcat:sex5apmc3zfcladgqnhp5he3xa
*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 ...##
###
Infinitesimal Thurston Rigidity and the Fatou-Shishikura Inequality
[article]

1999
*
arXiv
*
pre-print

We prove

arXiv:math/9902158v1
fatcat:qerdcgnkd5c3lf6r5xlu52y5yq
*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 ...##
###
I prefer pi: A brief history and anthology of articles in the American Mathematical Monthly (2015)
[chapter]

2016
*
Pi: The Next Generation
*

In celebration

doi:10.1007/978-3-319-32377-0_25
fatcat:w3za7435hnfgnaky4pekrfrr3y
*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. ...##
###
Principal solutions of recurrence relations and irrationality questions in number theory
[article]

2008
*
arXiv
*
pre-print

new series expansions

arXiv:math/0608577v2
fatcat:twl7qvhtbrgwzpsl4gkoyhepzy
*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) ). ...##
###
Apéry Limits: Experiments and Proofs
[article]

2020
*
arXiv
*
pre-print

An important component

arXiv:2011.03400v1
fatcat:5jvups4lwzdzrchoxb5e7yh2um
*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. ...##
###
Arithmetic hypergeometric series

2011
*
Russian Mathematical Surveys
*

Originally designed as

doi:10.1070/rm2011v066n02abeh004742
fatcat:o6c6ny5q5nbalpj6qreujrecdq
*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*). ...##
###
The Life of π: From Archimedes to ENIAC and Beyond
[chapter]

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. ...

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

2013
*
Acta Arithmetica
*

As an illustration, we give few examples

doi:10.4064/aa157-4-1
fatcat:ruouxaxoxzf7ljmgubzwjhondm
*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. ...##
###
Rational approximations to algebraic Laurent series with coefficients in a finite field
[article]

2011
*
arXiv
*
pre-print

As an illustration, we give few examples

arXiv:1102.5764v1
fatcat:niou6ac5azgkzjrgbysoe24h7m
*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. ...##
###
Apéry constants of homogeneous varieties
[article]

2016
*
arXiv
*
pre-print

We do numerical

arXiv:1604.04652v1
fatcat:knumot5lcnckticghjxgyddu7e
*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). ...##
###
Pseudo-unitary non-self-dual fusion categories of rank 4
[article]

2014
*
arXiv
*
pre-print

In doing so, we classify all

arXiv:1401.1879v2
fatcat:vkylxbtfrvaibhbukeejhvgaba
*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. ...
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