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 $2\times 2$ ratio theorem for the characteristic polynomial of the unitary group. We additionally identify arithmetic functions whose statistics mimic those of hook Schur functions.

22 pages. Incorporates referee suggestions

Cited by in corpus (2)

• Squarefree polynomials and Mobius values in short intervals and arithmetic progressions
• Arithmetic functions in short intervals and the symmetric group