On the mean value of the functions related to the divisor function on the ring of polynomials over a finite field

Abstract

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 $.