Les valeurs aux entiers pairs (strictement positifs) de la fonction $\zeta$ de Riemann sont transcendantes, car ce sont des multiples rationnels de puissances de $\pi$. En revanche, on sait très peu de choses sur la nature arithmétique des $\zeta (2k+1)$, pour $k\ge 1$ entier. Apéry a démontré en 1978 que $\zeta \left(3\right)$ est irrationnel. Rivoal a prouvé en 2000 qu’une infinité de $\zeta (2k+1)$ sont irrationnels, mais sans pouvoir en exhiber aucun autre que $\zeta \left(3\right)$. Il existe plusieurs points de vue sur la preuve d’Apéry ; celui des séries hypergéométriques...

We generalize a previous result due to Badea relating to the irrationality of some quick convergent infinite series.

The main result of this paper is a criterion for linear independence of continued fractions over the rational numbers. The proof is based on their special properties.

For $x\in ℂ$, $\left|x\right|\<1$, $s\in \mathbb{N}$, let ${\mathrm{Li}}_s\left(x\right)$ be the $s$-th polylogarithm of $x$. We prove that for any non-zero algebraic number $\alpha$ such that $\left|\alpha \right|\<1$, the $\mathbb{Q}\left(\alpha \right)$-vector space spanned by $1,{\mathrm{Li}}_1\left(\alpha \right),{\mathrm{Li}}_2\left(\alpha \right),\cdots$ has infinite dimension. This result extends a previous one by Rivoal for rational $\alpha$. The main tool is a method introduced by Fischler and Rivoal, which shows the coefficients of the polylogarithms in the relevant series to be the unique solution of a suitable Padé approximation problem.

Let $Q$ be a nonconstant polynomial with integer coefficients and without zeros at the non–negative integers. Essentially with the method of Hermite, a new proof is given on linear independence of values at rational points of the function$$\begin{array}{c}\hfill G\left(x\right)=\sum _{n=0}^{\infty }\frac{x^n}{Q\left(1\right)Q\left(2\right)\cdots Q\left(n\right)}.\end{array}$$

We consider the behavior of the power series ${}_0\left(z\right)={\sum }_{n=1}^{\infty }{\mu }^2\left(n\right)z^n$ as z tends to $e\left(\beta \right)=e^{2\pi i\beta }$ along a radius of the unit circle. If β is irrational with irrationality exponent 2 then ${}_0\left(e\left(\beta \right)r\right)=O\left({(1-r)}^{-1/2-\epsilon }\right)$. Also we consider the cases of higher irrationality exponent. We prove that for each δ there exist irrational numbers β such that ${}_0\left(e\left(\beta \right)r\right)=\Omega \left({(1-r)}^{-1+\delta }\right)$.