2024/01/31 by Luo, Qi, Ye, Yangbo
#11Y16 #11Y70 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2401.18082
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≤108 for 1≤ h≤1,000 are explored. Chowla's conjecture predicts that the conditional expectation of λ(n+h) given λ(n)=1 for 1≤ n≤ X converges to the conditional expectation of λ(n+h) given λ(n)=-1 for 1≤ n≤ X as X→∞. However, for finite X, these conditional expectations are different. The observed difference, together with the significant difference in χ2 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≤ h≤1,000, the study also tested the convergence speeds of Chowla's conjecture and found no relation on h.