Orthogonality of the M\"obius function to polynomials with applications to Linear Equations in Primes over $\mathbb{F}_p[x]$

Abstract

We prove that the M\"obius function is orthogonal to polynomials over $\mathbb{F}_q[x]$ (up to a characteristic condition). We use this orthogonality property to count prime solutions to affine-linear equations of bounded complexity in $\mathbb{F}_p[x]$, with analog to a work of Green and Tao.