A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of and , as well as to explain Rivoal’s recent result on infiniteness of irrational numbers in the set of odd zeta values, and to prove that at least one of the four numbers , , , and is irrational.
In this paper, we look at various arithmetic properties of the set of those positive integers n whose sum of digits in a fixed base b > 1 is a fixed positive integer s. For example, we prove that such integers can have many prime factors, that they are not very smooth, and that most such integers have a large prime factor dividing the value of their Euler φ function.
The Littlewood conjecture in Diophantine approximation claims thatholds for all real numbers and , where denotes the distance to the nearest integer. Its -adic analogue, formulated by de Mathan and Teulié in 2004, asserts thatholds for every real number and every prime number , where denotes the -adic absolute value normalized by . We survey the known results on these conjectures and highlight recent developments.
We present asymptotic representations for certain reciprocal sums of Fibonacci numbers and of Lucas numbers as a parameter tends to a critical value. As limiting cases of our results, we obtain Euler’s formulas for values of zeta functions.
Soit une suite strictement croissante d’entiers reconnaissable par un automate fini. Nous montrons qu’une condition nécessaire et suffisante pour que l’ensemble normal associé a soit exactement est que l’un au moins des sommets qui reconnaît la suite soit précédé dans le graphe de l’automate par un sommet possédant au moins deux circuits fermés distincts. Cette condition peut se traduire quantitativement en disant que la suite doit être plus “dense” que toute suite exponentielle.
We establish new combinatorial transcendence criteria for continued fraction expansions. Let be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients of is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton.
Nous dressons un rapide panorama de résultats allant dans le sens de la conjecture suivante : l’intersection d’une sous-variété d’une variété semi-abélienne et de l’union de tous les sous-groupes algébriques de de codimension au moins n’est pas Zariski-dense dans dès que n’est pas contenue dans un sous-groupe algébrique strict de .