The shifted convolution problem in function fields
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909505837596672 |
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| author | Florea, Alexandra Lalín, Matilde Malik, Amita Sahay, Anurag |
| author_facet | Florea, Alexandra Lalín, Matilde Malik, Amita Sahay, Anurag |
| contents | We study the shifted convolution problem for the divisor function in function fields in the large degree limit, that is, the average value of $d(f) d(f+h)$ where $f$ runs over monic polynomials in $\mathbb{F}_q[T]$ of a given degree, and $h$ is a given monic polynomial. We prove an asymptotic formula in the range $\operatorname{deg}(h) < (2-ε)\operatorname{deg}(f)$. We also consider mixed correlations and self-correlations of $r_χ= 1 \star χ$, the convolution of $1$ with a Dirichlet character mod $\ell$, where $\ell$ is a monic irreducible polynomial, proving asymptotic formulae in various ranges. This includes the case of quadratic characters, which yields results about correlations of norm-counting functions of quadratic extensions of $\mathbb{F}_q[T]$. A novel feature of our work is a Voronoi summation formula (equivalently, a functional equation for the Estermann function) in $\mathbb{F}_q[T]$ which was not previously available. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16067 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The shifted convolution problem in function fields Florea, Alexandra Lalín, Matilde Malik, Amita Sahay, Anurag Number Theory 11N37, 11R58, 11R59 We study the shifted convolution problem for the divisor function in function fields in the large degree limit, that is, the average value of $d(f) d(f+h)$ where $f$ runs over monic polynomials in $\mathbb{F}_q[T]$ of a given degree, and $h$ is a given monic polynomial. We prove an asymptotic formula in the range $\operatorname{deg}(h) < (2-ε)\operatorname{deg}(f)$. We also consider mixed correlations and self-correlations of $r_χ= 1 \star χ$, the convolution of $1$ with a Dirichlet character mod $\ell$, where $\ell$ is a monic irreducible polynomial, proving asymptotic formulae in various ranges. This includes the case of quadratic characters, which yields results about correlations of norm-counting functions of quadratic extensions of $\mathbb{F}_q[T]$. A novel feature of our work is a Voronoi summation formula (equivalently, a functional equation for the Estermann function) in $\mathbb{F}_q[T]$ which was not previously available. |
| title | The shifted convolution problem in function fields |
| topic | Number Theory 11N37, 11R58, 11R59 |
| url | https://arxiv.org/abs/2502.16067 |