Rudin-Shapiro function along irreducible polynomials over finite fields

Abstract

Let $q$ be an odd prime power and $\mathbb{F}_q$ be the finite field of $q$ elements. We define the Rudin-Shapiro function $R$ on monic polynomials $f=t^n+f_{n-1}t^{n-1}+\dots + f_0\in\mathbb{F}_q[t]$ over $\mathbb{F}_q$ by $$ R(f)=\sum_{i=1}^{n-1}f_if_{i-1}. $$ We investigate the distribution of the Rudin-Shapiro function along irreducible polynomials. We show that the number of irreducible polynomials $f$ with $R(f)=\gamma$ for any $\gamma\in\mathbb{F}_q$ is asymptotically $q^{n-1}/n$ as $n\rightarrow\infty$.