The summatory function of the Möbius function involving the greatest common divisor

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 $S_{\mu }(x):={\sum }_{mn\le x}\mu (\gcd (m,n))\mathrm{.}$ 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.