Correlation of arithmetic functions over $\mathbb{F}_q[T]$
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2018
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866916435568099328 |
|---|---|
| author | Gorodetsky, Ofir Sawin, Will |
| author_facet | Gorodetsky, Ofir Sawin, Will |
| contents | For a fixed polynomial $Δ$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+Δ$ are both irreducible, an $\mathbb F_q[T]$-analogue of the twin primes problem. In the large-$q$ limit, we obtain a lower-order term for this count if we consider non-monic polynomials, which depends on $Δ$ in a manner which is consistent with the Hardy-Littlewood Conjecture. We obtain a saving of $q$ if we consider monic polynomials only and $Δ$ is a scalar. To do this, we use symmetries of the problem to get for free a small amount of averaging in $Δ$. This allows us to obtain additional saving from equidistribution results for $L$-functions. We do all this in a combinatorial framework that applies to more general arithmetic functions than the indicator function of irreducibles, including the Möbius function and divisor functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1811_04834 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Correlation of arithmetic functions over $\mathbb{F}_q[T]$ Gorodetsky, Ofir Sawin, Will Number Theory For a fixed polynomial $Δ$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+Δ$ are both irreducible, an $\mathbb F_q[T]$-analogue of the twin primes problem. In the large-$q$ limit, we obtain a lower-order term for this count if we consider non-monic polynomials, which depends on $Δ$ in a manner which is consistent with the Hardy-Littlewood Conjecture. We obtain a saving of $q$ if we consider monic polynomials only and $Δ$ is a scalar. To do this, we use symmetries of the problem to get for free a small amount of averaging in $Δ$. This allows us to obtain additional saving from equidistribution results for $L$-functions. We do all this in a combinatorial framework that applies to more general arithmetic functions than the indicator function of irreducibles, including the Möbius function and divisor functions. |
| title | Correlation of arithmetic functions over $\mathbb{F}_q[T]$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/1811.04834 |