The covariance of almost-primes in $\mathbb{F}_q[T]$

Abstract: We estimate the covariance in counts of almost-primes in $\mathbb{F}_q[T]$, weighted by higher-order von Mangoldt functions. The answer takes a pleasant algebraic form. This generalizes recent work of Keating and Rudnick that estimates the variance of primes, and makes use, as theirs, of a recent equidistribution result of Katz. In an appendix we prove some related identities for random matrix statistics, which allows us to give a quick proof of a $2\times2$ ratio theorem for the characteristic polynomial of the unitary group. We additionally identify arithmetic functions whose statistics mimic those of hook Schur functions.