A definite recursive relation and some statistical properties for M\"obius function

Abstract

An elementary recursive relation for M$\ddot{\mathrm{o}}$bius function $\mu (n)$ is introduced by two simple ways. With this recursive relation, $\mu (n)$ can be calculated without directly knowing the factorization of the $n$. $\mu (1) \sim \mu (2 \times 10^7) $ are calculated recursively one by one. Based on these $2\times 10^7$ samples, the empirical probabilities of $\mu (n)$ of taking $-1$, 0, and 1 in classic statistics are calculated and compared with the theoretical probabilities in number theory. The numerical consistency between these two kinds of probability show that $\mu (n)$ could be seen as an independent random sequence when $n$ is large. The expectation and variance of the $\mu (n)$ are $0$ and $6 n/ \pi^2$, respectively. Furthermore, we show that any conjecture of the Mertens type is false in probability sense, and present an upper bound for cumulative sums of $\mu (n)$ with a certain probability.