Correlation of arithmetic functions over 𝔽𝑞[𝑇]
For a fixed polynomial 𝛥, we study the number of polynomials f of degree n over 𝔽𝑞 such that f and 𝑓+𝛥 are both irreducible, an 𝔽𝑞[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 m...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Springer
2019
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| _version_ | 1797055976194441216 |
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| author | Gorodetsky, O Sawin, W |
| author_facet | Gorodetsky, O Sawin, W |
| author_sort | Gorodetsky, O |
| collection | OXFORD |
| description | For a fixed polynomial 𝛥, we study the number of polynomials f of degree n over 𝔽𝑞 such that f and 𝑓+𝛥 are both irreducible, an 𝔽𝑞[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. |
| first_indexed | 2024-03-06T19:16:59Z |
| format | Journal article |
| id | oxford-uuid:18bdcc49-7284-403f-9f25-aa3467293d12 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T19:16:59Z |
| publishDate | 2019 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:18bdcc49-7284-403f-9f25-aa3467293d122022-03-26T10:44:52ZCorrelation of arithmetic functions over 𝔽𝑞[𝑇]Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:18bdcc49-7284-403f-9f25-aa3467293d12EnglishSymplectic ElementsSpringer2019Gorodetsky, OSawin, WFor a fixed polynomial 𝛥, we study the number of polynomials f of degree n over 𝔽𝑞 such that f and 𝑓+𝛥 are both irreducible, an 𝔽𝑞[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. |
| spellingShingle | Gorodetsky, O Sawin, W Correlation of arithmetic functions over 𝔽𝑞[𝑇] |
| title | Correlation of arithmetic functions over 𝔽𝑞[𝑇] |
| title_full | Correlation of arithmetic functions over 𝔽𝑞[𝑇] |
| title_fullStr | Correlation of arithmetic functions over 𝔽𝑞[𝑇] |
| title_full_unstemmed | Correlation of arithmetic functions over 𝔽𝑞[𝑇] |
| title_short | Correlation of arithmetic functions over 𝔽𝑞[𝑇] |
| title_sort | correlation of arithmetic functions over 𝔽𝑞 𝑇 |
| work_keys_str_mv | AT gorodetskyo correlationofarithmeticfunctionsoverfqt AT sawinw correlationofarithmeticfunctionsoverfqt |