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$$\begin{array}{c}\hfill G\left(x\right)=\sum _{n=0}^{\infty }\frac{x^n}{Q\left(1\right)Q\left(2\right)\cdots Q\left(n\right)}.\end{array}$$

Let $\mu \ge 2$ be a real number and let $ℳ\left(\mu \right)$ denote the set of real numbers approximable at order at least $\mu$ by rational numbers. More than eighty years ago, Jarník and, independently, Besicovitch established that the Hausdorff dimension of $ℳ\left(\mu \right)$ is equal to $2/\mu$. We investigate the size of the intersection of $ℳ\left(\mu \right)$ with Ahlfors regular compact subsets of the interval $[0,1]$. In particular, we propose a conjecture for the exact value of the dimension of $ℳ\left(\mu \right)$ intersected with the middle-third Cantor set and give several results...

We establish a new multiplicity lemma for solutions of a differential system extending Ramanujan’s classical differential relations. This result can be useful in the study of arithmetic properties of values of Riemann zeta function at odd positive integers (Nesterenko, 2011).

Depuis un peu plus de vingt ans, la recherche de minorations de combinaisons linéaires de logarithmes de nombres algébriques avec des coefficients algébriques a fait l'objet de nombreux travaux. Dès que le nombre de logarithmes dépasse 2, toutes les démonstrations utilisées jusqu'à présent reposaient sur la méthode de Baker. Nous proposons ici d'autres méthodes.

A criterion is given for studying (explicit) Baker type lower bounds of linear forms in numbers $1,{\Theta }_1,...,{\Theta }_m\in ℂ*$ over the ring $\mathbb{Z}$ of an imaginary quadratic field . This work deals with the simultaneous auxiliary functions case.

We prove that 7. 398 537 is an irrationality measure of $\zeta \left(2\right)={\pi }^2/6$. We employ double integrals of suitable rational functions invariant under a group of birational transformations of ${ℂ}^2$. The numerical results are obtained with the aid of a semi-infinite linear programming method.

Let ${\left(t_k\right)}_{k\ge 0}$ be the Thue–Morse sequence on $\{0,1\}$ defined by $t_0=0$, $t_{2k}=t_k$ and $t_{2k+1}=1-t_k$ for $k\ge 0$. Let $b\ge 2$ be an integer. We establish that the irrationality exponent of the Thue–Morse–Mahler number ${\sum }_{k\ge 0}{t}_k{b}^{-k}$ is equal to $2$.

Let ${\left(a_n\right)}_{n\ge 1}$ be the sequence given by $a_1=1$ and $a_n=n^{a_{n-1}}$ for $n\ge 2$. In this paper, we show that the only solution of the equation$$a_1+\cdots +a_n=m^l$$is in positive integers $l\>1,m$ and $n$ is $m=n=1$.

We give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field. Our proof is based on a method introduced in a different framework by Adamczewski and Cassaigne. It makes use of automata theory and, in our context, of a classical theorem due to Christol. We then introduce a new approach which allows us to strongly improve this general bound in many cases. As an illustration, we give a few examples of algebraic Laurent series for which...