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

TL;DR

This paper introduces a recursive relation for the Möbius function allowing calculation without factorization, and analyzes its statistical properties, showing it behaves like an independent random sequence for large n.

Contribution

It presents a new recursive method for computing the Möbius function and investigates its probabilistic behavior, providing empirical and theoretical insights.

Findings

Empirical probabilities of μ(n) match theoretical predictions.

μ(n) behaves like an independent random sequence for large n.

Provides bounds on cumulative sums of μ(n) with certain probability.

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.