Related papers: On simultaneous diophantine approximations to $\ze…

We report new hypergeometric constructions of rational approximations to Catalan's constant, $\log2$, and $\pi^2$, their connection with already known ones, and underlying "permutation group" structures. Our principal arithmetic achievement…

This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers $\zeta_{1},\zeta_{2},...,\zeta_{k}$. The approach relies on results on the connection between the set of all…

In 1978, Apery has given sequences of rational approximations to $\zeta(2)$ and $\zeta(3)$ yielding the irrationality of each of these numbers. One of the key ingredient of Apery's proof are second-order difference equations with polynomial…

We introduce a new exponent of simultaneous rational approximation $\widehat{\lambda}_{\min}(\xi,\eta)$ for pairs of real numbers $\xi,\eta$, in complement to the classical exponents $\lambda(\xi,\eta)$ of best approximation, and…

This note describes continued fraction representations for the rational approximations to the zeta function recently found by the author. It is tempting to think that these continued fractions might be analysed using a souped up version of…

In this paper we generalize Nesterenko's criterion to the case where the small linear forms have an oscillating behaviour (for instance given by the saddle point method). This criterion provides both a lower bound for the dimension of the…

A theorem of Kurzweil ('55) on inhomogeneous Diophantine approximation states that if $\theta$ is an irrational number, then the following are equivalent: (A) for every decreasing positive function $\psi$ such that $\sum_{q = 1}^\infty…

We revisit Beukers' modular-form proof of the irrationality of $\zeta(3)$ from the point of view of the auxiliary weight two modular form. For the Fricke group $\Gamma_0(6)^\star$, we show that Beukers' choice is not isolated: it belongs to…

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

We first recall the connection, going back to A. Thue, between rational approximation to algebraic numbers and integer solutions of some Diophantine equations. Next we recall the equivalence between several finiteness results on various…

Let $\alpha$ be a fixed quadratic irrational. Consider the Diophantine equation \[ y^a\ =\ q_{N_1} + \cdots + q_{N_K},\quad N_1 \geq \cdots \geq N_{K} \geq 0,\quad a, y \geq 2 \] where $(q_N)_{N\,\geq\,0}$ is the sequence of convergent…

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 Ap\'ery's Theorem…

We study properties of Diophantine exponents of lattices and so-called related "weak" uniform approximations introduced in recent papers by Oleg German, in the simplest two-dimensional case. In contrast to the multidimensional case, in the…

Using symbolic summation tools in the setting of difference rings, we prove a two-parametric identity that relates rational approximations to $\zeta(4)$.

An important component of Ap\'ery's proof that $\zeta (3)$ is irrational involves representing $\zeta (3)$ as the limit of the quotient of two rational solutions to a three-term recurrence. We present various approaches to such Ap\'ery…

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

In this note, using an idea from \cite{Amo-Carrillo-Sanchez} we derive some new series representations involving $\zeta(2n)$ and Euler numbers. Using a well-known series representation for the Clausen function, we also provide some new…

In this paper, we explore several threads arising from our recent joint work on arithmetic holonomy bounds, which were originally devised to prove new irrationality results based on the method of Ap\'ery limits. We propose a new method to…

In this paper we study the three-dimensional analogue of the relation between the irrationality exponent of a real number and the growth of its regular continued fraction partial quotients. As a multidimensional generalisation of continued…

We prove a result on approximations to a real number $\theta$ by algebraic numbers of degree $\le 2$ in the case when we have information about the uniform Diophantine exponent $\hat{\omega}$ for the linear form $x_0 +\theta…