Let $ \mathbb{ F}_q[T]$\, be the ring of polynomials over a finite field $ \mathbb{ F}_q $. Let $ g: \mathbb{ F}_q[T] \rightarrow \mathbb{ R} $ be a multiplicative function such that for any irreducible polynomial $ P $ over $ \mathbb{ F}_q $ and any $ k \ge 1 $, the equality $ d_k = g (P ^ k) $ holds for some arbitrary sequence of reals $\{ d_k\}_{ k=1}^{ \infty}$. In this paper, we get an explicit formula for the sum $$ T (N) = \sum\limits_{ \substack{ \deg F=N F \text{ is monic}}}{ g (F)}, $$ and also derive different asymptotics when this sum in cases of $ q \to \infty; \ q \to \infty, \ N \to \infty; \ q ^ N \to \infty $.
Categories: math.NT