arXiv2022
In this article we study the asymptotic behaviour of the correlation functions over polynomial ring $\mathbb{F}_q[x]$. Let $\mathcal{M}_{n, q}$ and $\mathcal{P}_{n, q}$ be the set of all monic polynomials and monic irreducible polynomials of degree $n$ over $\mathbb{F}_q$ respectively. For multiplicative functions $ψ_1$ and $ψ_2$ on $\mathbb{F}_q[x]$, we obtain asymptotic formula for the following correlation functions for a fixed $q$ and $n\to \infty$ \begin{align*} &S_{2}(n, q):=\displaystyle\sum_{f\in \mathcal{M}_{n, q}}ψ_1(f+h_1) ψ_2(f+h_2), \\ &R_2(n, q):=\displaystyle\sum_{P\in \mathcal{P}_{n, q}}ψ_1(P+h_1)ψ_2(P+h_2), \end{align*} where $h_1, h_2$ are fixed polynomials of degree $<n$ over $\mathbb{F}_q$. As a consequence, for real valued additive functions $\tilde{ψ_1}$ and $\tilde{ψ_2}$ on $\mathbb{F}_q[x]$ we show that for a fixed $q$ and $n\to \infty$, the following distribution functions \begin{align*} &\frac{1}{|\mathcal{M}_{n, q}|}\Big|\{f\in \mathcal{M}_{n, q} : \tilde{ψ_1}(f+h_1)+\tilde{ψ_2}(f+h_2)\leq x\}\Big|,\\ & \frac{1}{|\mathcal{P}_{n, q}|}\Big|\{P\in \mathcal{P}_{n, q} : \tilde{ψ_1}(P+h_1)+\tilde{ψ_2}(P+h_2)\leq x\}\Big| \end{align*} converges weakly towards a limit distribution.