q-Apery Irrationality Proofs by q-WZ Pairs

An elementary proof of Apery's theorem

We present a new `elementary' proof of the irrationality of $ζ(3)$ based on some recent `hypergeometric' ideas of Yu.Nesterenko, T.Rivoal, and K.Ball, and on Zeilberger's algorithm of creative telescoping.

Irrationalité de valeurs de zêta (d'après Apéry, Rivoal, ...)

This survey text deals with irrationality, and linear independence over the rationals, of values at positive odd integers of Riemann zeta function. The first section gives all known proofs (and connections between them) of Apéry's Theorem (1978) : $ζ(3)$ is irrational. The second section is devoted to a variant of the proof, published by Rivoal and Ball-Rivoal, that infinitely many $ζ(2n+1)$ are irrational. The end of this text deals with more quantitative statements.

Experimenting with Apery Limits and WZ pairs

This article, dedicated with admiration in memory of Jon and Peter Borwein, illustrates by example, the power of experimental mathematics, so dear to them both, by experimenting with so-called Apery limits and WZ pairs. In particular we prove a weaker form of an intriguing conjecture of Marc Chamberland and Armin Straub (in an article dedicated to Jon Borwein), and generate lots of new Apery limits. We also rediscovered an infinite family of cubic irrationalities, that suggested very good effective irrationality measures (lower than Liouville's generic 3), and that seemed to go down to the opt…

Irrationality proofs for zeta values, moduli spaces and dinner parties

A simple geometric construction on the moduli spaces $\mathcal{M}_{0,n}$ of curves of genus $0$ with $n$ ordered marked points is described which gives a common framework for many irrationality proofs for zeta values. This construction yields Apéry's approximations to $ζ(2)$ and $ζ(3)$, and for larger $n$, an infinite family of small linear forms in multiple zeta values with an interesting algebraic structure. It also contains a generalisation of the linear forms used by Ball and Rivoal to prove that infinitely many odd zeta values are irrational.

Proof of some conjectures of Z.-W. Sun on congruences for Apery polynomials

The Apery polynomials are defined by $A_n(x)=\sum_{k=0}^{n}{n\choose k}^2{n+k\choose k}^2 x^k$ for all nonnegative integers $n$. We confirm several conjectures of Z.-W. Sun on the congruences for the sum $\sum_{k=0}^{n-1}(-1)^k(2k+1) A_k(x)$ with $x\in Z$.

Automatic Discovery of Irrationality Proofs and Irrationality Measures

We illustrate the power of Experimental Mathematics and Symbolic Computation to suggest irrationality proofs of natural constants, and the determination of their irrationality measures. Sometimes such proofs can be fully automated, but sometimes there is still need for a human touch.

Congruences for sporadic sequences and modular forms for non-congruence subgroups

In the course of the proof of the irrationality of zeta(2) R. Apery introduced numbers b_n = \sum_{k=0}^n {n \choose k}^2{n+k \choose k}. Stienstra and Beukers showed that for the prime p > 3 Apery numbers satisfy congruence b((p-1)/2) = 4a^2-2p mod p, if p = a^2+b^2 (where a is odd). Later, Zagier found some generalizations of Apery numbers, so called sporadic sequences, and recently Osburn and Straub proved similar congruences for all but one of the six Zagier's sporadic sequences (three cases were already known to be true) and conjectured the congruence for the sixth sequence. In this paper…

关于 $ζ_2(5)$ 无理性的一个注记

A note on the irrationality of $ζ_2(5)$

受 Apéry 证明 $ζ(3)$ 无理性的启发，构造有理逼近序列证明 $ζ_2(5)$ 的无理性，并给出其无理性测度的上界。

General proof for irrationality of infinite sums based on Fourier's proof

In this paper we review a general proof for the irrationality property of numbers which take a certain form of infinite sums.

Sequences, modular forms and cellular integrals

It is well-known that the Apéry sequences which arise in the irrationality proofs for $ζ(2)$ and $ζ(3)$ satisfy many intriguing arithmetic properties and are related to the $p$th Fourier coefficients of modular forms. In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.

ζ(3)无理性的模形式证明

A Modular Form Proof of the Irrationality of $ζ\left(3\right)$

该研究利用6级模形式，通过构造艾希勒积分得到特定幂级数，依据贝克斯无理性准则，证明了ζ(3)的无理性。

Sun's Three Conjectures on Apéry-like Sums Involving Harmonic Numbers

In this paper, we will give another proof of Zhi-Wei Sun's three conjectures on Apéry-like sums involving harmonic numbers by proving some identities among special values of multiple polylogarithms.

模形式与ζ(3)有理近似数值探索

Modular Forms and Numerical Explorations of Rational Approximations to $ζ(3)$

通过辅助权重二模形式重新审视Beukers对ζ(3)无理性的证明，展示其属于一个参数仿射族，并应用于其他零亏格Fricke群。

A Geometric Proof to Cantor's Theorem and an Irrationality Measure for Some Cantor's Series

Generalizing a geometric idea due to J. Sondow, we give a geometric proof for the Cantor's Theorem. Moreover, it is given an irrationality measure for some Cantor series.

Hermite 风格的无理性证明

Irrationality proofs à la Hermite

本文通过研读 Hermite 1873 年的论文，追溯了 Niven 证明 π 无理性的积分起源，揭示了 Hermite 对 π² 无理性的简单证明，并给出了 r tan r 及某些 Bessel 函数比值无理性的新证明。

类似Beukers的证明用于ζ(2)和ζ(3)的无理数性

Beukers-like proofs of irrationality for $ζ{(2)}$ and $ζ{(3)}$

本文提供ζ(2)和ζ(3)无理数性的分步证明，改进了Beukers方法，有助于理解及扩展至更高ζ值、Catalan常数等。

Motivic irrationality proofs

We exhibit geometric conditions on a family of toric hypersurfaces under which the value of a canonical normal function at a point of maximal unipotent monodromy is irrational.

Recurrent proofs of the irrationality of certain trigonometric values

We use recurrences of integrals to give new and elementary proofs of the irrationality of pi, tan(r) for all nonzero rational r, and cos(r) for all nonzero rational r^2. Immediate consequences to other values of the elementary transcendental functions are also discussed.

On an incomplete argument of Erdos on the irrationality of Lambert series

We show that the Lambert series $f(x)=\sum d(n) x^n$ is irrational at $x=1/b$ for negative integers $b < -1$ using an elementary proof that finishes an incomplete proof of Erdos.

An elementary proof of the irrationality of Tschakaloff series

We present a new proof of the irrationality of values of the series $T_q(z)=\sum_{n=0}^\infty z^nq^{-n(n-1)/2}$ in both qualitative and quantitative forms. The proof is based on a hypergeometric construction of rational approximations to $T_q(z)$.

The Irrationality Measure of Pi is at most 7.103205334137...

We use a variant of Salikhov's ingenious proof that the irrationality measure of $π$ is at most $7.606308\dots$ to prove that, in fact, it is at most $7.103205334137\dots$. Accompanying Maple package: While this article has a fully rigorous human-made and human-readable proof of the claim in the title, it was discovered thanks to the Maple package available from http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/pimeas.html

A simpler proof of a Katsurada's theorem and rapidly converging series for $ζ{(2n+1)}$ and $β{(2n)}$

In a recent work on Euler-type formulae for even Dirichlet beta values, i.e. $β{(2n)}$, I have derived an exact closed-form expression for a class of zeta series. From this result, I have conjectured closed-form summations for two families of zeta series. Here in this work, I begin by using a known formula by Wilton to prove those conjectures. As example of applications, some special cases are explored, yielding rapidly converging series representations for the Apéry constant, $ζ(3)$, and the Catalan constant, $G = β(2)$. Interestingly, our series for $\,ζ(3)\,$ converges faster than that used…

Primes, Pi, and Irrationality Measure

A folklore proof of Euclid's theorem on the infinitude of primes uses the Euler product and the irrationality of $ζ(2) = π^2/6$. A quantified form of Euclid's Theorem is Bertrand's postulate $p_{n+1} < 2p_n$. By quantifying the folklore proof using an irrationality measure for $6/π^2$, we give a proof (communicated to Paulo Ribenboim in 2005) of a much weaker upper bound on $p_{n+1}$.

在 Lean 4 中证明 $ζ(3)$ 的无理性的形式化证明

A Formal Proof of the Irrationality of $ζ(3)$ in Lean 4

在 Lean 4 中，使用 Beukers 方法并扩展 Mathlib 及形式化素数定理，首次形式化证明了 $ζ(3)$ 的无理性。

A way to prove the irrationality of Zeta(4)

The proof of the irrationality of Zeta(5) is a long standing open problem, but here only the case of Zeta(4) = (Pi^4)/90 is considered. The present paper suggests an approach for the irrationality of Zeta(4) along the lines of those known for proving the irrationality of Zeta(3).

Irrationality proof of a $q$-extension of $ζ(2)$ using little $q$-Jacobi polynomials

We show how one can use Hermite-Padé approximation and little $q$-Jacobi polynomials to construct rational approximants for $ζ_q(2)$. These numbers are $q$-analogues of the well known $ζ(2)$. Here $q=\frac{1}{p}$, with $p$ an integer greater than one. These approximants are good enough to show the irrationality of $ζ_q(2)$ and they allow us to calculate an upper bound for its measure of irrationality: $μ(ζ_q(2))\leq 10π^2/(5π^2-24) \approx 3.8936$. This is sharper than the upper bound given by Zudilin (\textit{On the irrationality measure for a $q$-analogue of $ζ(2)$}, Mat. Sb. \textbf{193} (2…

Irrationality proof of certain Lambert series using little q-Jacobi polynomials

We apply the Pade technique to find rational approximations to % \[h^{\pm}(q_1,q_2)=\sum_{k=1}^\infty\frac{\q_1^k}{1\pm \q_2^k}, 0<q_1,q_2<1, q_1\in\mathbb{Q}, q_2=1/p_2, p_2\in\mathbb{N}\setminus\{1\}.\] % A separate section is dedicated to the special case $q_i=q^{r_i}, r_i\in\mathbb{N}, q=1/p, p\in\mathbb{N}\setminus\{1\}$. In this construction we make use of little $q$-Jacobi polynomials. Our rational approximations are good enough to prove the irrationality of $h^{\pm}(q_1,q_2)$ and give an upper bound for the irrationality measure.

Note On the Irrationality of the L-Function Constants L(s, X)

A unified proof of the irrationality of the special values L(n, X), n > 1 an integer, of the beta L-function is put forward in this note. The first case of n = 2 seems to confirm that the Catalan constant L(2, X) is an irrational number.

Proof of the Tijdeman-Zagier Conjecture via Slope Irrationality and Term Coprimality

The Tijdeman-Zagier conjecture states no integer solution exists for $A^X+B^Y=C^Z$ with positive integer bases and integer exponents greater than 2 unless gcd$(A,B,C)>1$. Any set of values that satisfy the conjecture correspond to a lattice point on a Cartesian graph which subtends a line in multi-dimensional space with the origin. Properties of the slopes of these lines in each plane are established as a function of coprimality of terms, such as irrationality, which enable us to explicitly prove the conjecture by contradiction.

A geometric proof that $e$ is irrational and a new measure of its irrationality

We give a simple geometric proof that $e$ is irrational, using a construction of a nested sequence of closed intervals with intersection $e$. The proof leads to a new measure of irrationality for $e$: if $p$ and $q$ are integers with $q > 1$, then $|e - p/q| > 1/(S(q)+1)!$, where $S(q)$ is the smallest positive integer such that $S(q)!$ is a multiple of $q$. We relate this measure for $e$ to a known one and to the greatest prime factor of an integer. We make two conjectures and recall a theorem of Cantor that can be proved by a similar construction.

Irrationality of some p-adic L-values

We give a proof of the irrationality of the $p$-adic zeta-values $ζ_p(k)$ for $p=2,3$ and $k=2,3$. Such results were recently obtained by F.Calegari as an application of overconvergent $p$-adic modular forms. In this paper we present an approach using classical continued fractions discovered by Stieltjes. In addition we show irrationality of some other $p$-adic $L$-series values, and values of the $p$-adic Hurwitz zeta-function.

Shidlovsky's multiplicity estimate and Irrationality of zeta values

In this paper we follow the approach of Bertrand-Beukers (and of later work of Bertrand), based on differential Galois theory, to prove a very general version of Shidlovsky's lemma that applies to Pad{é} approximation problems at several points, both at functional and numerical levels (i.e., before and after evaluating at a specific point). This allows us to obtain a new proof of the Ball-Rivoal theorem on irrationality of infinitely many values of Riemann zeta function at odd integers, inspired by the proof of the Siegel-Shidlovsky theorem on values of E-functions: Shidlovsky's lemma is used …

A Hypergeometric Approach, Via Linear Forms Involving Logarithms, to Irrationality Criteria for Euler's Constant

Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler's constant $γ$. The proof is by reduction to known irrationality criteria for $γ$ involving a Beukers-type double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linear forms in 1, $γ$, and logarithms from Nesterenko-type series of rational functions. In the Appendix, Sergey Zlobin gives a change-of-variables proof that the series and the double integral are equal.

Irrationality exponent and rational approximations with prescribed growth

Let $ξ$ be a real irrational number. We are interested in sequences of linear forms in 1 and $ξ$, with integer coefficients, which tend to 0. Does such a sequence exist such that the linear forms are small (with given rate of decrease) and the coefficients have some given rate of growth? If these rates are essentially geometric, a necessary condition for such a sequence to exist is that the linear forms are not too small, a condition which can be expressed precisely using the irrationality exponent of $ξ$. We prove that this condition is actually sufficient, even for arbitrary rates of growth …

$ζ(5)$ is irrational

We present an elementary proof of the irrationality of $ζ(5)$ based upon the Dirichlet's approximation theorem and the Prime Number Theorem.

Metric Diophantine approximation with respect to planar distance functions

We outline a proof of an analogue of Khintchine's Theorem in R^2, where the ordinary height is replaced by a distance function satisfying an irrationality condition as well as certain decay and symmetry conditions.

特殊函数 $e^{x}E_{1}(x)$ 的有理函数逼近及其在 Euler-Gompertz 常数 $δ$ 的无理性中的应用

Rational function approximations of the special function $e^{x}E_{1}(x)$ and applications to irrationality of Euler-Gompertz constant $δ$

本文提出一种构造特殊函数 $e^{x}E_{1}(x)$ 连分数逼近的方法，证明了其收敛性，并以此构造逼近 Euler-Gompertz 常数 $δ$ 的有理数列，给出了证明其无理性的充分条件。