  Let $\gcd(m,n)$ denote the greatest common divisor of the positive integers
$m$ and $n$, and let $\mu$ represent the M\" obius function. For any real
number $x>5$, we define the summatory function of the M\" obius function
involving the greatest common divisor as $ S_{\mu}(x) := \sum_{mn\leq x}
\mu(\gcd(m,n)). $ In this paper, we present an asymptotic formula for
$S_{\mu}(x)$. Assuming the Riemann Hypothesis, we delve further into the
asymptotic behavior of $S_{\mu}(x)$ and derive a mean square estimate for its
error term. Our proof employs the Perron formula, Parseval's theorem, complex
integration techniques, and the properties of the Riemann zeta-function.
