Fractional part correlations and the Möbius function
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929421073514496 |
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| author | Chavez, Gordon |
| author_facet | Chavez, Gordon |
| contents | We show that $$ \sum_{n\neq m}\frac{μ(n)μ(m)}{nm}E_{X}\left(\{nx\}\{mx\}\right)=-\frac{9}{2π^{2}}+O\left(\frac{1}{X}\right), $$ where $x$ is uniformly distributed in $[0,X]$ with $X\in \mathbb{N}$, $E_{X}(.)$ denotes the expected value, $μ(.)$ denotes the Möbius function, and $\{.\}$ denotes the fractional part function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09343 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractional part correlations and the Möbius function Chavez, Gordon Number Theory 11A25, 11K65, 11N64 We show that $$ \sum_{n\neq m}\frac{μ(n)μ(m)}{nm}E_{X}\left(\{nx\}\{mx\}\right)=-\frac{9}{2π^{2}}+O\left(\frac{1}{X}\right), $$ where $x$ is uniformly distributed in $[0,X]$ with $X\in \mathbb{N}$, $E_{X}(.)$ denotes the expected value, $μ(.)$ denotes the Möbius function, and $\{.\}$ denotes the fractional part function. |
| title | Fractional part correlations and the Möbius function |
| topic | Number Theory 11A25, 11K65, 11N64 |
| url | https://arxiv.org/abs/2402.09343 |