Correlation of multiplicative functions over function fields

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}}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 $\widetilde{{\psi }_1}$ and $\widetilde{{\psi }_2}$ 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{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.

24 pages; Comments are welcome

References in corpus (1)

• The autocorrelation of the Möbius function and Chowla's conjecture for the rational function field in characteristic 2

Cited by in corpus (1)

• Correlations of multiplicative functions in function fields