Correlation of multiplicative functions over Fq[x]$\mathbb {F}_q[x]$: A pretentious approach

Abstract

Let denote the set of monic polynomials of degree n over a finite field of q elements. For multiplicative functions , using the recently developed “pretentious method,” we establish a “local‐global” principle for correlation functions of the form where are fixed polynomials. These results were then applied to find limiting distributions of sums of additive functions. We also study correlations with Dirichlet characters and Hayes characters. As a consequence, we give a new proof of a function field analog of Kátai's conjecture that states that if the average of the first divided difference of a completely multiplicative function whose values lie on the unit circle is zero, then it must be a Hayes character. We further extend this result to pairs and triplets of completely multiplicative functions.