Correlation of multiplicative functions over Fq[x]: A pretentious approach

Let M-n denote the set of monic polynomials of degree n over a finite field F-q of q elements. For multiplicative functions psi(1), psi(2), using the recently developed "pretentious method," we establish a "local-global" principle for correlation functions of the form Sigma(f is an element of Mn) psi(1)(f + h(1))psi(2)(f + h(2)) as n -> infinity (q remains fixed), where h(1), h(2) is an element of F-q[x] 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 Katai'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.