Rudin-Shapiro function along irreducible polynomials over finite fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911103111397376 |
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| author | Mérai, László |
| author_facet | Mérai, László |
| contents | 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 $R(f)=γ$ for any $γ\in\mathbb{F}_q$ is asymptotically $q^{n-1}/n$ as $n\rightarrow\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19012 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rudin-Shapiro function along irreducible polynomials over finite fields Mérai, László Number Theory 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 $R(f)=γ$ for any $γ\in\mathbb{F}_q$ is asymptotically $q^{n-1}/n$ as $n\rightarrow\infty$. |
| title | Rudin-Shapiro function along irreducible polynomials over finite fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.19012 |