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 $\le x$.