We give a combinatorial interpretation for the positive moments of the values at the edge of the critical strip of the $L$-functions of modular forms of $GL\left(2\right)$ and $GL\left(3\right)$. We deduce some results about the asymptotics of these moments. We extend this interpretation to the moments twisted by the eigenvalues of Hecke operators.

En admettant la conjecture de Dickson, nous démontrons que, pour chaque couple d’entiers $q\>0$ et $k\>0$, il existe une partie infinie $L_{q,k}\subset \mathbb{N}$ telle que, pour chacun des entiers $n\in L_{q,k}$ et tout entier $s$ tel que $0\<\left|s\right|\le q$, on ait $n+s=\left|s\right|t_1...t_k$ où $t_1\<...\<t_k$ sont des nombres premiers. De même, pour chaque couple d’entiers $q\>0$ et $k\>0$, il existe une partie infinie $M_{q,k}\subset \mathbb{N}$ telle que, pour chacun des entiers $n\in M_{q,k}$ et tout entier $s$ (nul ou non ) de l’intervalle $\left[-q,q\right]$, on ait $n+s=l{t}_1...t_k$ où $t_1\<...\<t_k$ sont des nombres premiers et l’entier $l$ appartient à l’intervalle $\left[1,2q+1\right]$. La lecture non standard...

Let $F\left(X\right)={\sum }_{n\ge 0}{(-1)}^{\epsilon ₙ}{X}^{-\lambda ₙ}$ be a real lacunary formal power series, where εₙ = 0,1 and ${\lambda }_{n+1}/\lambda ₙ>2$. It is known that the denominators Qₙ(X) of the convergents of its continued fraction expansion are polynomials with coefficients 0, ±1, and that the number of nonzero terms in Qₙ(X) is the nth term of the Stern-Brocot sequence. We show that replacing the index n by any 2-adic integer ω makes sense. We prove that $Q_{\omega }\left(X\right)$ is a polynomial if and only if ω ∈ ℤ. In all the other cases $Q_{\omega }\left(X\right)$ is an infinite formal power series; we discuss its algebraic...

Dans un corps fini, toute série formelle algébrique en une indéterminée est la diagonale d'une fraction rationnelle en deux indéterminées (Furstenberg 67). Dans cet article, nous donnons une nouvelle preuve de ce résultat, par des méthodes purement combinatoires.

We obtain new results concerning the Lang-Trotter conjectures on Frobenius traces and Frobenius fields over single and double parametric families of elliptic curves. We also obtain similar results with respect to the Sato-Tate conjecture. In particular, we improve a result of A. C. Cojocaru and the second author (2008) towards the Lang-Trotter conjecture on average for polynomially parameterised families of elliptic curves when the parameter runs through a set of rational numbers of bounded height....

This paper consists of three parts. In the first part we prove a general theorem on the image of a language $K$ under a substitution, in the second we apply this to the special case when $K$ is the language of balanced words and in the third part we deal with recurrent Z-words of minimal block growth.

Large families of pseudorandom binary sequences and lattices are constructed by using the multiplicative inverse and estimates of exponential sums in a finite field. Pseudorandom measures of binary sequences and lattices are studied.

We prove that there is a small but fixed positive integer $ϵ$ such that for every prime $p$ larger than a fixed integer, every subset $S$ of the integers modulo $p$ which satisfies $\left|2S\right|\le (2+ϵ)\left|S\right|$ and $2\left(\right|2S\left|\right)-2\left|S\right|+3\le p$ is contained in an arithmetic progression of length $\left|2S\right|-\left|S\right|+1$. This is the first result of this nature which places no unnecessary restrictions on the size of $S$.

Nous étudions une loi de composition sur les carrés magiques, qui a déjà été introduite dans la littérature, qui munit l'ensemble de tous les carrés magiques d'une structure de semi-groupe (monoïde). Nous prouvons ensuite une conjecture de Adler et Li, ce semi-groupe est libre.