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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.18136 |
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| _version_ | 1866911314961498112 |
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| author | Dieguez, Alexandre |
| author_facet | Dieguez, Alexandre |
| contents | For a fixed irrational $θ> 0$ with a prescribed irrationality measure function, we study the correlation $\int_1^X Δ(x) Δ(θx) dx$, where $Δ$ is the Dirichlet error term in the divisor problem. When $θ$ has a finite irrationality measure, it is known that decorrelation occurs at a rate expressible in terms of this measure. Strong decorrelation occurs for all positive irrationals, except possibly Liouville numbers. We show that for irrationals with a prescribed irrationality measure function $ψ$, decorrelation can be quantified in terms of $ψ^{-1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18136 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On certain correlations into the divisor problem Dieguez, Alexandre Number Theory For a fixed irrational $θ> 0$ with a prescribed irrationality measure function, we study the correlation $\int_1^X Δ(x) Δ(θx) dx$, where $Δ$ is the Dirichlet error term in the divisor problem. When $θ$ has a finite irrationality measure, it is known that decorrelation occurs at a rate expressible in terms of this measure. Strong decorrelation occurs for all positive irrationals, except possibly Liouville numbers. We show that for irrationals with a prescribed irrationality measure function $ψ$, decorrelation can be quantified in terms of $ψ^{-1}$. |
| title | On certain correlations into the divisor problem |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.18136 |