Waring's problem with almost proportional summands
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912113073586176 |
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| author | Rakhmonov, Zarullo Rakhmonov, Firuz |
| author_facet | Rakhmonov, Zarullo Rakhmonov, Firuz |
| contents | For $n \geq 3$, an asymptotic formula is derived for the number of representations of a sufficiently large natural number $N$ as a sum of $r = 2^n + 1$ summands, each of which is an $n$-th power of natural numbers $x_i$, $i = \overline{1, r}$, satisfying the conditions $$
|x_i^n-μ_iN|\le H,\qquad H\ge N^{1-θ(n,r)+\varepsilon},\qquad θ(n,r)=\frac2{(r+1)(n^2-n)}, $$ where $μ_1, \ldots, μ_r$ are positive fixed numbers, and $μ_1 + \ldots + μ_n = 1$. This result strengthens the theorem of E.M.Wright. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06153 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Waring's problem with almost proportional summands Rakhmonov, Zarullo Rakhmonov, Firuz Number Theory For $n \geq 3$, an asymptotic formula is derived for the number of representations of a sufficiently large natural number $N$ as a sum of $r = 2^n + 1$ summands, each of which is an $n$-th power of natural numbers $x_i$, $i = \overline{1, r}$, satisfying the conditions $$ |x_i^n-μ_iN|\le H,\qquad H\ge N^{1-θ(n,r)+\varepsilon},\qquad θ(n,r)=\frac2{(r+1)(n^2-n)}, $$ where $μ_1, \ldots, μ_r$ are positive fixed numbers, and $μ_1 + \ldots + μ_n = 1$. This result strengthens the theorem of E.M.Wright. |
| title | Waring's problem with almost proportional summands |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.06153 |