Correlation of multiplicative functions over function fields (1905.09303v2)

Abstract: 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 $\psi_1$ and $\psi_2$ 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}}\psi_1(f+h_1) \psi_2(f+h_2), \ &R_2(n, q):=\displaystyle\sum{P\in \mathcal{P}{n, q}}\psi_1(P+h_1)\psi_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 $\tilde{\psi_1}$ and $\tilde{\psi_2}$ 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{\psi_1}(f+h_1)+\tilde{\psi_2}(f+h_2)\leq x}\Big|,\ & \frac{1}{|\mathcal{P}{n, q}|}\Big|{P\in \mathcal{P}_{n, q} : \tilde{\psi_1}(P+h_1)+\tilde{\psi_2}(P+h_2)\leq x}\Big| \end{align*} converges weakly towards a limit distribution.