Algorithms of the Möbius function by random forests and neural networks

The Mobius function mu(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu (n)$$\end{document} is known for containing limited information on the prime factorization of n. Its known algorithms, however, are all based on factorization and hence are exponentially slow on logn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log n$$\end{document}. Consequently, a faster algorithm of mu(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu (n)$$\end{document} could potentially lead to a fast algorithm of prime factorization which in turn would throw doubt upon the security of most public-key cryptosystems. This research introduces novel approaches to compute mu(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu (n)$$\end{document} using random forests and neural networks, harnessing the additive properties of mu(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu (n)$$\end{document}. The machine learning models are trained on a substantial dataset with 317,284 observations (80%), comprising five feature variables, including values of n within the range of 4x109\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\times 10<^>9$$\end{document}. We implement the Random Forest with Random Inputs (RFRI) and Feedforward Neural Network (FNN) architectures. The RFRI model achieves a predictive accuracy of 0.9493, a recall of 0.5865, and a precision of 0.6626. On the other hand, the FNN model attains a predictive accuracy of 0.7871, a recall of 0.9477, and a precision of 0.2784. These results strongly support the effectiveness and validity of the proposed algorithms.