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

## TL;DR

This paper derives explicit formulas and asymptotic behaviors for sums of multiplicative functions related to the divisor function over polynomial rings in finite fields, extending understanding of their mean values in various limits.

## Contribution

It provides explicit formulas and asymptotic results for sums of multiplicative functions over polynomials in finite fields, generalizing divisor function analysis.

## Key findings

- Explicit formula for sum T(N) over polynomials of degree N
- Asymptotic behaviors as q, N, or q^N tend to infinity
- Generalization of divisor function mean value results

## 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 $.