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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.07456 |
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| _version_ | 1866912590381187072 |
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| author | Liu, Haomin Lü, Jiadong Xie, Yonghao |
| author_facet | Liu, Haomin Lü, Jiadong Xie, Yonghao |
| contents | We give an affirmative answer to a question asked by N. Moshchevitin \cite{m1} in his lecture at International Congress of Basic Science, Beijing, 2024 (see also \cite{m}, Section 6.3). The question is that whether the remainder $$ R_n=\sum_{j=1}^{2^n}\left(ξ_{j,n}-\frac{j}{2^n}\right)^2-2^n\int_0^1( ?(x)-x))^2\text{d}x $$ tends to $0$ when $n$ tends to infinity, where $ξ_{j,n}$ are elements of the Stern-Brocot sequence and $?(x)$ denotes Minkowski Question-Mark Function. We present some extended results and give a correct proof of a theorem on the Fourier-Stieltjes coefficient of the inverse function of $?(x)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sums with Stern-Brocot sequences and Minkowski question mark function Liu, Haomin Lü, Jiadong Xie, Yonghao Number Theory We give an affirmative answer to a question asked by N. Moshchevitin \cite{m1} in his lecture at International Congress of Basic Science, Beijing, 2024 (see also \cite{m}, Section 6.3). The question is that whether the remainder $$ R_n=\sum_{j=1}^{2^n}\left(ξ_{j,n}-\frac{j}{2^n}\right)^2-2^n\int_0^1( ?(x)-x))^2\text{d}x $$ tends to $0$ when $n$ tends to infinity, where $ξ_{j,n}$ are elements of the Stern-Brocot sequence and $?(x)$ denotes Minkowski Question-Mark Function. We present some extended results and give a correct proof of a theorem on the Fourier-Stieltjes coefficient of the inverse function of $?(x)$. |
| title | Sums with Stern-Brocot sequences and Minkowski question mark function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.07456 |