Distribution of neighboring values of the Liouville and Möbius functions
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913217417052160 |
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| author | Luo, Qi Ye, Yangbo |
| author_facet | Luo, Qi Ye, Yangbo |
| contents | Let $λ(n)$ and $μ(n)$ denote the Liouville function and the Möbius function, respectively. In this study, relationships between the values of $λ(n)$ and $λ(n+h)$ up to $n\leq10^8$ for $1\leq h\leq1,000$ are explored. Chowla's conjecture predicts that the conditional expectation of $λ(n+h)$ given $λ(n)=1$ for $1\leq n\leq X$ converges to the conditional expectation of $λ(n+h)$ given $λ(n)=-1$ for $1\leq n\leq X$ as $X\rightarrow\infty$. However, for finite $X$, these conditional expectations are different. The observed difference, together with the significant difference in $χ^2$ tests of independence, reveals hidden additive properties among the values of the Liouville function. Similarly, such additive structures for $μ(n)$ for square-free $n$'s are identified. These findings pave the way for developing possible, and hopefully efficient, additive algorithms for these functions. The potential existence of fast, additive algorithms for $λ(n)$ and $μ(n)$ may eventually provide scientific evidence supporting the belief that prime factorization of large integers should not be too difficult. For $1\leq h\leq1,000$, the study also tested the convergence speeds of Chowla's conjecture and found no relation on $h$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_18082 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distribution of neighboring values of the Liouville and Möbius functions Luo, Qi Ye, Yangbo Number Theory 11Y16, 11Y70 Let $λ(n)$ and $μ(n)$ denote the Liouville function and the Möbius function, respectively. In this study, relationships between the values of $λ(n)$ and $λ(n+h)$ up to $n\leq10^8$ for $1\leq h\leq1,000$ are explored. Chowla's conjecture predicts that the conditional expectation of $λ(n+h)$ given $λ(n)=1$ for $1\leq n\leq X$ converges to the conditional expectation of $λ(n+h)$ given $λ(n)=-1$ for $1\leq n\leq X$ as $X\rightarrow\infty$. However, for finite $X$, these conditional expectations are different. The observed difference, together with the significant difference in $χ^2$ tests of independence, reveals hidden additive properties among the values of the Liouville function. Similarly, such additive structures for $μ(n)$ for square-free $n$'s are identified. These findings pave the way for developing possible, and hopefully efficient, additive algorithms for these functions. The potential existence of fast, additive algorithms for $λ(n)$ and $μ(n)$ may eventually provide scientific evidence supporting the belief that prime factorization of large integers should not be too difficult. For $1\leq h\leq1,000$, the study also tested the convergence speeds of Chowla's conjecture and found no relation on $h$. |
| title | Distribution of neighboring values of the Liouville and Möbius functions |
| topic | Number Theory 11Y16, 11Y70 |
| url | https://arxiv.org/abs/2401.18082 |