Consider the group ${SL}_2\left({\mathbf{O}}_K\right)$ over the ring of algebraic integers of a number field $K$. Define the height of a matrix to be the maximum over all the conjugates of its entries in absolute value. Let ${SL}_2({\mathbf{O}}_K,t)$ be the number of matrices in ${SL}_2\left({\mathbf{O}}_K\right)$ with height bounded by $t$. We determine the asymptotic behaviour of ${SL}_2({\mathbf{O}}_K,t)$ as $t$ goes to infinity including an error term,$${SL}_2({\mathbf{O}}_K,t)=C{t}^{2n}+O\left(t^{2n-\eta }\right)$$with $n$ being the degree of $K$. The constant $C$ involves the discriminant of $K$, an integral depending only on the signature of $K$, and the value of the Dedekind zeta function...

In this paper we generalize the method used to prove the Prime Number Theorem to deal with finite fields, and prove the following theorem: $$\pi \left(x\right)=\frac{q}{q-1}\frac{x}{{log}_{q}x}+\frac{q}{{(q-1)}^2}\frac{x}{{log}_q^{2}x}+O\left(\frac{x}{{log}_q^{3}x}\right),x=q^n\to \infty$$ where $\pi \left(x\right)$ denotes the number of monic irreducible polynomials in $F_q\left[t\right]$ with norm $\le x$.

Hasse showed the existence and computed the Dirichlet density of the set of primes $p$ for which the order of $2(modp)$ is odd; it is $7/24$. Here we mimic successfully Hasse’s method to compute the density ${\delta }_q$ of monic irreducibles $P$ in ${𝔽}_q\left[X\right]$ for which the order of $X(modP)$ is odd. But on the way, we are also led to a new and elementary proof of these densities. More observations are made, and averages are considered, in particular, an average of the ${\delta }_p$’s as $p$ varies through all rational primes.

On montre à l’aide de méthodes de crible, de méthodes issues de la théorie des formes automorphes et de géométrie algébrique ainsi qu’à l’aide de la loi de Sato-Tate verticale que le signe des sommes de Kloosterman $\mathrm{Kl}(1,1;n)$ change une infinité de fois pour $n$ parcourant les entiers sans facteur carré ayant au plus $18$ facteurs premiers. Ceci améliore un résultat précédent de Fouvry et Michel qui avaient obtenu $23$ à la place de $18$.