Automorphisms of polynomial algebras and Dirichlet series

Let Fq[x,y] be the polynomial algebra in two variables over the finite field Fq with q elements. We give an exact formula and the asymptotics for the number pn of automorphisms (f,g) of Fq[x,y] such that max{deg(f),deg(g)}=n. We describe also the Dirichlet series generating functionp(s)=∑n⩾1pnns. The same results hold for the automorphisms of the free associative algebra Fq〈x,y〉. We have also obtained analogues for free algebras with two generators in Nielsen-Schreier varieties of algebras.