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 $f=t^n+f_{n-1}{t}^{n-1}+\cdots +f_0\in {\mathbb{F}}_q[t]$ over ${\mathbb{F}}_q$ by $$R(f)=\sum _{i=1}^{n-1}{f}_i{f}_{i-1}.$$ 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 $n\to \infty$.