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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2412.00723 |
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| _version_ | 1866915042442608640 |
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| author | Qiu, Yuan Kalmynin, Alexander B. |
| author_facet | Qiu, Yuan Kalmynin, Alexander B. |
| contents | The research in the subfield of analytic number theory around error term of summation of sigma functions possesses a history which can be dated back to the mid-19th century when Dirichlet provided an $O(\sqrt{n})$ estimation of error term of summation of $d(n)$. Later, G. Voronoi, G. Kolesnik, and M.N. Huxley (to name just a few) contributed more on the upper bound on the error term of summation of sigma functions. As for $Ω$-theorems, G.H. Hardy was the first contributor. Later researchers on this topic include G.H. Hardy and T.H. Gronwall, but the amount of academic effort is much sparser than $O$-theorems. This research aims to provide a better $Ω$-bound for the error term of summation of fractional sigma function $σ_α(n)$ on the range $0 < α< \frac{1}{2}$, obtaining the result $Ω((x \ln x)^{\frac{1}{4}+\fracα{2}})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00723 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Omega theorem for fractional sigma function Qiu, Yuan Kalmynin, Alexander B. Number Theory The research in the subfield of analytic number theory around error term of summation of sigma functions possesses a history which can be dated back to the mid-19th century when Dirichlet provided an $O(\sqrt{n})$ estimation of error term of summation of $d(n)$. Later, G. Voronoi, G. Kolesnik, and M.N. Huxley (to name just a few) contributed more on the upper bound on the error term of summation of sigma functions. As for $Ω$-theorems, G.H. Hardy was the first contributor. Later researchers on this topic include G.H. Hardy and T.H. Gronwall, but the amount of academic effort is much sparser than $O$-theorems. This research aims to provide a better $Ω$-bound for the error term of summation of fractional sigma function $σ_α(n)$ on the range $0 < α< \frac{1}{2}$, obtaining the result $Ω((x \ln x)^{\frac{1}{4}+\fracα{2}})$. |
| title | Omega theorem for fractional sigma function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2412.00723 |