Rudin-Shapiro function along irreducible polynomials over finite fields

TL;DR

This paper studies the distribution of the Rudin-Shapiro function evaluated on irreducible polynomials over finite fields, showing that for large degrees, the values are evenly distributed among all possible outputs.

Contribution

It establishes asymptotic formulas for the number of irreducible polynomials with a given Rudin-Shapiro value over finite fields.

Findings

Number of irreducible polynomials with fixed R value is approximately q^{n-1}/n.

Distribution of R values among irreducible polynomials is uniform asymptotically.

Provides insight into polynomial value distributions 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 $n\rightarrow\infty$.