Wadim Zudilin
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.
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Using WZ forms, Apery-style proofs of the irrationality of the q-analogues of the Harmonic seires and Ln(2) are given. For the q-analogue of Ln(2), this method of proof produces an improved irrationality measure.
A simplification of Apéry's proof of the irrationality of ζ(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. Conclusive evidence is also presented that these constructions arise from a continued fraction due to Ramanujan.
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 $x\in Z$.
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.
Roger Apery's seminal method for proving irrationality is "turned on its head" and taught to computers, enabling a one second redux of the original proof of zeta(3), and many new irrationality proofs of many new constants, alas, none of them is both famous and not-yet-proved-irrational.
In this paper, we will apply the ideas from the mirror symmetry of Calabi-Yau threefolds to study the modular forms and one-parameter family of K3 surfaces found by Beukers and Peters, which provide enlightenment to the two mysterious sequences constructed by Apéry in his proof of the irrationality of $ζ(3)$. We will construct a fourth order differential operator and a prepotential from the canonical solutions of this differential operator. The third derivative of this prepotential with respect to the mirror map defines a Yukawa coupling that is a weight-4 modular form. The instanton expansion…
There are only aleph-zero rational numbers, while there are 2 to the power aleph-zero real numbers. Hence the probability that a randomly chosen real number would be rational is 0. Yet proving rigorously that any specific, natural, real constant, is irrational is usually very hard, witness that there are still no proofs of the irrationality of the Euler-Mascheroni constant, the Catalan constant, or Zeta(5). Inspired by Frits Beukers' elegant rendition of Apery's seminal proofs of the irrationality of Zeta(2) and Zeta(3), and heavily using algorithmic proof theory, we systematically searched fo…
It is known that the numbers which occur in Apery's proof of the irrationality of zeta(2) have many interesting congruence properties while the associated generating function satisfies a second order differential equation. We prove supercongruences for a generalization of numbers which arise in Beukers' and Zagier's study of integral solutions of Apery-like differential equations.
Proof of some congruence conjectures of Z.-H. Sun involving Apéry-like numbers
该研究证明了孙智宏提出的一个涉及类Apéry数的素数模p⁴同余猜想，明确了含Franel数的求和式与Euler数的同余关系。
This monograph is intended to be considered as my habilitation (D.Sc.) thesis; because of that and as everything has already appeared in English, it is performed exclusively in Russian. The monograph comprises a detailed introduction and seven chapters that represent part of my work influenced by Apéry's proof from 1978 of the irrationality of $ζ(2)$ and $ζ(3)$, the values of Riemann's zeta function. Chapter 1 is about "at least one of the four numbers $ζ(5)$, $ζ(7)$, $ζ(9)$ and $ζ(11)$ is irrational" (based in part on arXiv:math.NT/0206176). Chapter 2 explains a connection between the general…
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…
Following Apery's proof of the irrationality of zeta(3), Beukers found an elegant reinterpretation of Apery's arguments using modular forms. We show how Beukers arguments can be adapted to a p-adic setting. In this context, certain functional equations arising from Eichler integrals are replaced by the notion of overconvergent p-adic modular forms, and the periods themselves arise not as coefficients of period polynomials but as constant terms of p-adic Eisenstein series. We prove that the analogue of zeta(3) is irrational for p = 2 and 3, as well as the 2-adic analogue of Catalan's constant.
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.
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 $T_q(z)$.
We present an elementary proof for Ljunggren equation
An elementary gap in the proof of corollary 2.2 was found, the claim in the first version of the paper is thus retracted.
We derive the mean square of the divisor function using only elementary techniques.
An elementary proof of the Selberg identity for Kloosterman sums
本文利用加性特征的正交性，给出了Kloosterman和的Selberg恒等式的初等证明。
In 1930, A.A.K. Ayyangar allegedly produced the missing proof that the ancient Indian Chakravala algorithm, designed to solve Pell's equation, always halts. Refining his own elementary arguments, we give a correct and shorter proof.
We give an elementary proof of the Mazur-Tate-Teitelbaum conjecture for elliptic curves by using Kato's element.
We present a very elementary proof of the uniqueness of Markoff numbers which are prime powers or twice prime powers, in the sense that it uses neither algebraic number theory nor hyperbolic geometry.
A New Elementary Proof of Landau's Prime Ideal Theorem, and Associated Results
本文给出Landau素数理想定理的新初等证明，推广了Richter的素数定理证明，并包含素数理想计数函数等分布的相关结果。
The main goal of this article is to present an elementary proof of Ramanujan's identity for odd zeta values. Our proof solely relies on a Mittag-Leffler type expansion for hyperbolic cotangent function and Euler's identity for even zeta values.
We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.
We give a short proof of the following known congruence: for every odd prime $p$ $$\sum_{k=0}^{p-1}{2k\choose k}^2 16^{-k}\equiv (-1)^{p-1\over 2}\pmod{p^2}.$$ Moreover, we provide some new results connected with the above congruence.
Mordell in 1958 gave a new proof of the three squares theorem. Those techniques were generalized by Blackwell, et al., in 2016 to characterize the integers represented by the remaining six "Ramanujan-Dickson ternaries". We continue the generalization of these techniques to four additional forms.
An elementary proof of some Ramanujan-type identities
本文通过初等方法证明了将黎曼ζ函数在整数点的平方表示为涉及双曲函数、digamma函数和伯努利数等级数的恒等式，并修正了文本中的不准确之处和参考文献。
An elementary formulation of Kubota's proof of Satz 4 about biquadratic number fields
本文旨在提供Kubota关于双二次数域Satz 4证明的初等表述，因为该命题是计算Polya群阶的基石。
An elementary proof of the local Kronecker-Weber theorem
本文给出了局部Kronecker-Weber定理的一个新的初等、自包含且显式的证明，仅使用离散赋值理论和本科代数课程中的工具。
For a two parameter family of Bernoulli numbers $B_{n, p}$ the exponential generating function is derived by elementary methods.
An Elementary Proof of the Generalization of the Binet Formula for $k$-bonacci Numbers
本研究提出了$k$-bonacci数Binet公式推广形式的初等证明，该公式通过$k$-bonacci递推的特征多项式根实现数的闭式计算，为相关数论问题提供了基础证明方法。
An elementary proof of the Artin--Springer theorem for generalized quadratic forms over quaternion algebras
本文为四元数除代数上各向异性广义二次型在奇数次域扩张下保持各向异性提供了初等证明。
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.
An elementary algebraic proof of the fundamental theorem of algebra
本文给出代数基本定理的一个新的初等代数证明，并推广了Joseph Shipman近期得到的一个更一般的版本，关键思想是理解多项式获得给定次数因子的扩域。
An elementary proof of Cohen-Gabber theorem in the equal characteristic $p>0$ case
本文提供了一个新的初等证明，用于Cohen-Gabber定理在等特征$p>0$情况下的应用。
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