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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.  ...  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]

Krishnan Rajkumar
2012 arXiv   pre-print
A simplification 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

M. Prévost
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.  ...

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.  ...  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]

Jaroslav Hančl, Simon Kristensen
2018 arXiv   pre-print
Finally, specialising 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]

N. A. Carella
2019 arXiv   pre-print
Furthermore, assuming 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)

Masayoshi Hata
2000 Acta Arithmetica
A new 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  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]

J. Arvesú, A. Soria-Lorente
2012 arXiv   pre-print
New (infinitely many) rational approximants to \zeta(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)

Georges Rhin, Carlo Viola
2001 Acta Arithmetica
In his proof of the irrationality of ζ(3), Apéry  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]

Tewodros Amdeberhan
1998 arXiv   pre-print
Using WZ pairs we present an infinite family 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 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]

F. M. S. Lima
2016 arXiv   pre-print
In this note, I develop step-by-step proofs 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)

Jorge Arvesú, Anier Soria-Lorente
2019 Mathematics
These solutions constitute 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  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]

Cyril Furtlehner, Stéphane Ouvry
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]

Masato Kobayashi
2021 arXiv   pre-print
We show four new integral representations for $\zeta(3)$ 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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