Probabilistic interpretation of the M\"obius function identity and the Riemann Hypothesis

TL;DR

This paper explores the probabilistic behavior of the Möbius function and its implications for the Riemann Hypothesis, suggesting that statistical properties of integers support the hypothesis.

Contribution

It introduces a probabilistic interpretation of the Möbius function and provides new statistical insights into the distribution of squarefree integers, supporting the Riemann Hypothesis.

Findings

Asymptotic densities of squarefree integers are 8/π² (odd) and 4/π² (even).

Statistics of Möbius function outcomes resemble coin flips for large squarefree numbers.

Results support the plausibility of the Riemann Hypothesis based on integer statistics.

Abstract

We obtained the probabilities for the values of the M\"obius function for arbitrary numbers and found that the asymptotic densities of the squarefree integers among the odd and even numbers are $8/\pi^2$ and $4/\pi^2$, respectively. It is determined that statistics of successive outcomes of the M\"obius function for very large squarefree odd and even numbers behaves similar to statistics of heads and tails of two flipping coins. These preliminary results are giving arguments supporting the Riemann Hypothesis. Its plausibility is based on statistical phenomena for integers.