On the mean value of functions related to the divisors function in the ring of polynomials over a finite field

Abstract

Let $ g: \mathbb{F}_q[T] \rightarrow \mathbb{R} $ be a multiplicative function which values at the degrees of the irreducible polynomial, depends only on the exponent, such that $g(P^k)=d_k$ polynomial $P$ and for some arbitrary sequence of reals $\{d_k\}_{k=1}^{\infty}$. This paper regards the sum $$ T (N) = \sum\limits_{\substack{\deg F=N \\ F \text{ is monic}}}{g (F)}, $$ where $ F $ ranges over polynomials of degree $ N $ with leading coefficient equal to 1 (unitary polynomials). For the sum $ T (N) $, an exact formula is found, and various asymptotics are calculated in cases of \\ $ q \to \infty; \ q \to \infty, \ N \to \infty; \ q ^ N \to \infty $. In particular, the following asymptotic formulas are obtained
$$\sum\limits_{\substack{\deg F=N \\ F \text{ is monic}}}\tau(F^k)=\binom{k+N}{N}q^N+O_{N,k}\left(q^{N-1}\right),\ \ N\ge 1,\ q\to\infty;
$$
$$
\sum\limits_{\substack{\deg F=N \\ F \text{ is monic}}}\dfrac{1}{\tau(F)}=\dfrac{q^N}{4^N}\left(\binom{2N}{N}-\dfrac{2}{3}\binom{2N-4}{N-2}q^{-1}+O\left(\ \dfrac{4^N}{\sqrt{N}}q^{-2}\right)\right),\ N\to\infty,\ q\to\infty;
$$
$$\sum\limits_{\substack{\deg F=N \\ F \text{ is monic}}}\dfrac{1}{\tau(F)}=C_1\cdot\dfrac{\binom{2N}{N}}{4^N}q^N+O\left(\dfrac{q^{N-0.5}}{N^{1.5}}\right),\ \ C_1=\prod_{l=1}^{+\infty}\left(\sqrt{q^{2l}-q^{l}}\ln\dfrac{q^l}{q^l-1}\right)^{\pi_q(l)},\ q^N\to\infty;$$
where $\tau(F)$ is a number of monic divisors of $F$, and $\pi_q(l)$ is a number of monic irreducible polynomials of degree $l$.
The second and third equalities are analogous for polynomials over a finite field of one of Ramanujan's results
$$\sum_{n\leq x}{\dfrac{1}{d(n)}}=\dfrac{x}{\sqrt{\ln x}}\left(a_0+\dfrac{a_1}{\ln{x}}+\ldots+\dfrac{a_N}{(\ln{x})^N}+O_N\left(\dfrac{1}{(\ln{x})^{N+1}}\right)\right),$$
where $d(n)$ is a classical divisor function, and $a_i$ are some constants. In particular,
$$a_0=\dfrac{1}{\sqrt{\pi}}\prod\limits_{p}\ln\dfrac{p}{p-1}\sqrt{p(p-1)}.$$