  Let GF(q)[x,y] be the polynomial algebra in two variables over the finite
field GF(q) with q elements. We give an exact formula and the asymptotics for
the number p(n) of automorphisms (f,g) of GF(q)[x,y] such that
max{deg(f),deg(g)}=n. We describe also the Dirichlet series generating function
p(1)/1^s+p(2)/2^s+p(3)/3^s+.... The same results hold for the automorphisms of
the free associative algebra GF(q)<x,y>. We have also obtained analogues for
free algebras with two generators in Nielsen - Schreier varieties of algebras.
