Correlation of multiplicative functions over function fields

TL;DR

This paper investigates the asymptotic behavior of correlation functions of multiplicative and additive functions over polynomial rings in finite fields, providing formulas and distribution convergence results as degree n tends to infinity.

Contribution

It derives asymptotic formulas for correlation functions of multiplicative functions over polynomial rings and establishes distribution convergence for additive functions in this setting.

Findings

Asymptotic formulas for correlation functions over polynomial rings.

Distribution functions of additive functions converge weakly as degree n increases.

Results hold for fixed finite field size q and large degree n.

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.