Asymptotic and non-asymptotic results for a binary additive problem involving Piatetski-Shapiro numbers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911970360295424 |
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| author | Yoshida, Yuuya |
| author_facet | Yoshida, Yuuya |
| contents | For all $α_1,α_2\in(1,2)$ with $1/α_1+1/α_2>5/3$, we show that the number of pairs $(n_1,n_2)$ of positive integers with $N=\lfloor{n_1^{α_1}}\rfloor+\lfloor{n_2^{α_2}}\rfloor$ is equal to $Γ(1+1/α_1)Γ(1+1/α_2)Γ(1/α_1+1/α_2)^{-1}N^{1/α_1+1/α_2-1} + o(N^{1/α_1+1/α_2-1})$ as $N\to\infty$, where $Γ$ denotes the gamma function. Moreover, we show a non-asymptotic result for the same counting problem when $α_1,α_2\in(1,2)$ lie in a larger range than the above. Finally, we give some asymptotic formulas for similar counting problems in a heuristic way. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_16691 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic and non-asymptotic results for a binary additive problem involving Piatetski-Shapiro numbers Yoshida, Yuuya Number Theory Primary 11D85 11D04 11D72 11L07, Secondary 11B30 11B25 For all $α_1,α_2\in(1,2)$ with $1/α_1+1/α_2>5/3$, we show that the number of pairs $(n_1,n_2)$ of positive integers with $N=\lfloor{n_1^{α_1}}\rfloor+\lfloor{n_2^{α_2}}\rfloor$ is equal to $Γ(1+1/α_1)Γ(1+1/α_2)Γ(1/α_1+1/α_2)^{-1}N^{1/α_1+1/α_2-1} + o(N^{1/α_1+1/α_2-1})$ as $N\to\infty$, where $Γ$ denotes the gamma function. Moreover, we show a non-asymptotic result for the same counting problem when $α_1,α_2\in(1,2)$ lie in a larger range than the above. Finally, we give some asymptotic formulas for similar counting problems in a heuristic way. |
| title | Asymptotic and non-asymptotic results for a binary additive problem involving Piatetski-Shapiro numbers |
| topic | Number Theory Primary 11D85 11D04 11D72 11L07, Secondary 11B30 11B25 |
| url | https://arxiv.org/abs/2403.16691 |