Liouville函数与Möbius函数的相邻值分布
Distribution of neighboring values of the Liouville and Möbius functions
AI总结：
本研究通过计算10^8范围内、间隔1到1000的相邻值，发现Liouville函数与无平方因子n对应的Möbius函数存在隐藏加性结构，验证了有限范围下Chowla猜想的条件期望差异，且收敛速度与间隔h无关，为开发两类函数的加性算法、支撑大整数素因子分解难度的相关观点提供了依据。
摘要：
设$\lambda(n)$和$\mu(n)$分别表示Liouville函数与Möbius函数。本研究探究了在$n\leq10^8$范围内、$1\leq h\leq1000$时$\lambda(n)$与$\lambda(n+h)$的取值关联。Chowla猜想预测，当$X\rightarrow\infty$时，在$1\leq n\leq X$范围内，给定$\lambda(n)=1$时$\lambda(n+h)$的条件期望，会收敛到给定$\lambda(n)=-1$时$\lambda(n+h)$的条件期望。但对于有限的$X$，这些条件期望存在差异。观测到的差异，加上独立性$\chi^2$检验中显现的显著差异，揭示了Liouville函数取值间隐藏的加性性质。类似地，研究也在无平方因子的$n$对应的$\mu(n)$中发现了这类加性结构。这些发现为开发针对这些函数的可行、且有望是高效的加性算法铺平了道路。$\lambda(n)$和$\mu(n)$的快速加性算法的潜在存在，最终可能为“大整数的素因子分解不应过于困难”这一观点提供科学依据。对于$1\leq h\leq1000$，本研究还测试了Chowla猜想的收敛速度，发现其与$h$无关。
英文摘要：
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 $h$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
