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

2021
*
arXiv
*
pre-print

This paper presents a complete formal verification

arXiv:1912.06611v6
fatcat:kijni3uvvjcpjhvm4esnq3bcgy
*of*a proof that*the*evaluation*of**the*Riemann zeta function at*3*is irrational, using*the*Coq proof assistant. ...*The*rest*of**the*proof combines arithmetical ingredients and asymptotic analysis, which we conduct by extending*the*Mathematical Components libraries. ... We also thank Cyril Cohen, Pierre Roux and Enrico Tassi for their help, in particular with*the*libraries this work depends on. ...##
###
A simplification of Apéry's proof of the irrationality of ζ(3)
[article]

2012
*
arXiv
*
pre-print

A simplification

arXiv:1212.5881v1
fatcat:pc6a6s5735emvbodkqstu5ngjy
*of*Ap\'ery's proof*of**the**irrationality**of*\zeta(*3*) is presented. ...*The*construction*of*approximations is motivated from*the*viewpoint*of*2-dimensional recurrence relations which simplifies many*of**the*details*of**the*proof. ... Waldschmidt for their suggestions and encouragement during various stages*of*this project. ...##
###
A new proof of the irrationality of ζ(2) and ζ(3) using Padé approximants

1996
*
Journal of Computational and Applied Mathematics
*

*the*

*irrationality*

*of*((2) and

*of*((

*3*). ...

*The*same method applied to

*the*partial sums

*of*ln(1 -t) also proves

*the*

*irrationality*

*of*certain values

*of*

*the*logarithm. ... Huttner (Lille I) for

*the*reference to Ap6ry's paper. I am grateful to Prof. C. Brezinski for his pertinent comments. ...

##
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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. ...*The*formalization*of*this proof is complete up to a weak corollary*of**the*Prime Number Theorem. ... Therefore if*ζ*(*3*) were a rational number, then ℓ n*ζ*(*3*), and hence σ n = 2ℓ*3*n (a n*ζ*(*3*) − b n ), would be an integer for n larger than*the*denominator*of**ζ*(*3*) . ...##
###
Metrical irrationality results related to values of the Riemann ζ-function
[article]

2018
*
arXiv
*
pre-print

Finally, specialising

arXiv:1802.03946v1
fatcat:4t63i6hjabevroqkuhanabxwo4
*the*criteria used, we give some new criteria for*the**irrationality**of*$\zeta(k)$,*the*Euler--Mascheroni constant and*the*latter two series. ... We introduce a one-parameter family*of*series associated to*the*Riemann $\zeta$-function and prove that*the*values*of**the*elements*of*this family at integers are linearly independent over*the*rationals ... Then,*the*set {γ,*ζ*(2),*ζ*(*3*), . . . ,*ζ*(K)} ...##
###
The Zeta Quotient ζ(3)/ π^3 is Irrational
[article]

2019
*
arXiv
*
pre-print

Furthermore, assuming

arXiv:1906.10618v2
fatcat:aesq4mwbuzhyrlsiell5wrl7ym
*the**irrationality**of**the*second odd zeta value $\zeta(5)$, it is shown that $\zeta(5)\ne r\pi^5$ for any rational number $r \in \mathbb{Q}$. ... This note proves that*the*first odd zeta value does not have a closed form formula $\zeta(*3*)\ne r \pi^*3*$ for any rational number $r \in \mathbb{Q}$. ... As an illustration*of**the*versatility*of*this technique, conditional on*the**irrationality**of**the*zeta value*ζ*(5), in Theorem 5.5 it is shown that*ζ*(5)/π 5 is irrational. ...##
###
A new irrationality measure for ζ(3)

2000
*
Acta Arithmetica
*

A new

doi:10.4064/aa-92-1-47-57
fatcat:vdam7snf4jcs7ci3eprs6azsrm
*irrationality*measure for*ζ*(*3*) by Masayoshi Hata (Kyoto) 4 log( √ 2 + 1) −*3*= 13.4178202 . . . ...*The*author would like to thank*the*referee for suggesting many improvements and correcting some minor errors. ... Various proofs*of**the**irrationality**of**ζ*(*3*) are known. V. N. Sorokin [9] constructed a number*of*Hermite-Padé approximations to certain series which lead to*the**irrationality**of**ζ*(*3*). Yu. V. ...##
###
On new rational approximants to ζ(3)
[article]

2012
*
arXiv
*
pre-print

New (infinitely many) rational approximants to \zeta(

arXiv:1204.6712v1
fatcat:zhr2jyyi2vehdlc5f345duwhda
*3*) proving its*irrationality*are given. ... A comparison*of*our approximants with Ap\'ery's approximants to \zeta(*3*) is shown. ...*The*above new rational approximants (2.10) prove*the**irrationality**of**ζ*(*3*). Proof. ...##
###
The group structure for ζ(3)

2001
*
Acta Arithmetica
*

In his proof

doi:10.4064/aa97-3-6
fatcat:wvw6epjuwbhvfn2hbfvuzen53q
*of**the**irrationality**of**ζ*(*3*), Apéry [1] gave sequences*of*rational approximations to*ζ*(2) = π 2 /6 and to*ζ*(*3*) yielding*the**irrationality*measures µ(*ζ*(2)) < 11.85078 . . . and µ(*ζ*(*3*)) ...*The**irrationality*measure*of**ζ*(*3*). ... For E we choose*the*set*of**the*following 15 permutations: ϕϑ 2 ϕ, ϑϕϑ 2 ϕ, ϑ*3*ϕϑ 2 ϕ, ϑ 4 ϕϑ 2 ϕ, ϑ 5 ϕϑ 2 ϕ, ϑ 7 ϕϑ 2 ϕ, ϕϑ 6 ϕ, ϑϕϑ 6 ϕ, ϑ 4 ϕϑ 6 ϕ, ϑ 5 ϕϑ 6 ϕ, χϑ*3*χ, ϑχϑ*3*χ, ϑ 2 χϑ*3*χ, ϑ*3*χϑ*3*...##
###
Faster and Faster convergent series for ζ(3)
[article]

1998
*
arXiv
*
pre-print

Using WZ pairs we present an infinite family

arXiv:math/9804126v1
fatcat:55tm7vvaujhufktzbbjrxoyopi
*of*accelerated series for computing $\zeta(*3*)$. ... AMS Subject Classification: Primary 05A Alf van der Poorten [P] gave a delightful account*of*Apéry's proof [A]*of**the**irrationality**of**ζ*(*3*). ... In this paper, after choosing a particular F (where its companion G is then produced by*the*amazing Maple package EKHAD accompanying [PWZ] ), we will give a list*of*accelerated series calculating*ζ*(*3*) ...##
###
On simultaneous diophantine approximations to ζ(2) and ζ(3)

2014
*
Journal of Number Theory
*

We present a hypergeometric construction

doi:10.1016/j.jnt.2014.06.007
fatcat:h5ywstr625c43hut723rgkcny4
*of*rational approximations to $\zeta(2)$ and $\zeta(*3*)$ which allows one to demonstrate simultaneously*the**irrationality**of*each*of**the*zeta values, as well as ... Finally,*the*properties*of*this newer concept are studied and linked to*the*classical*irrationality*exponent and its generalisations given recently by S. Fischler. ... Apéry was*the*first to establish*the**irrationality**of*such*ζ*(s): he proved [Apé79] in 1978 that*ζ*(*3*) is irrational. ...##
###
Beukers-like proofs of irrationality for ζ(2) and ζ(3)
[article]

2016
*
arXiv
*
pre-print

In this note, I develop step-by-step proofs

arXiv:1308.2720v2
fatcat:tzatjym765flbjy5xbewthsaje
*of**irrationality*for $\,\zeta{(2)}\,$ and $\,\zeta{(*3*)}$. ... this kind*of*proof and/or trying to extend it to higher zeta values, Catalan's constant, or other related numbers. ... In this note, I develop rigorous Beukers-like proofs*of**irrationality*for*ζ*(2) and*ζ*(*3*), in full details. ...##
###
On Infinitely Many Rational Approximants to ζ(3)

2019
*
Mathematics
*

These solutions constitute

doi:10.3390/math7121176
fatcat:cuwmnikfrncphgmnllqhfhn3lq
*the*numerator and denominator sequences*of*rational approximants to*ζ*(*3*) . ... A correspondence from*the*set*of*parameters involved in*the*holonomic difference equation to*the*set*of*holonomic bi-sequences formed by these numerators and denominators appears. ... Introduction Apéry's proof [1]*of**the**irrationality**of**the*number*ζ*(*3*) (where*ζ*(s) = ∑ ∞ n=1 n −s , Re s > 1) is based on*the*second order difference equation (three-term recurrence relation) α n y n ...##
###
Integrals involving four Macdonald functions and their relation to 7zeta(3)/2
[article]

2004
*
arXiv
*
pre-print

*The*latter are shown to be related to 7zeta(

*3*)/2. A generalization to 2n integration variables is given which yields only zeta at odd arguments. ... A family

*of*multiple integrals over four variables is rewritten in terms

*of*a family

*of*simple integrals involving

*the*product

*of*four modified Bessel (Macdonald functions). ...

*the*

*irrationality*

*of*

*ζ*(

*3*). ...

##
###
Integral representations for ζ(3) with the inverse sine function
[article]

2021
*
arXiv
*
pre-print

We show four new integral representations for $\zeta(

arXiv:2108.01247v1
fatcat:2hsxndqwfvfmplj3grrkdicdnq
*3*)$ as a reformulation*of*Ewell (1990) and Yue-Williams (1993) with*the*inverse sine function and Wallis integral. ... As a consequence, we also show a local integral representation for*the*trilogarithm function. ...*The*author would like to thank*the*organizer Shigeru Iitaka, Yuji Yamaga and Kouichi Nakagawa for fruitful discussions. ...
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