The summatory function of the M\"obius function involving the greatest common divisor

Let (m,n) denote the greatest common divisor of the positive integers m and n, and let μ 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 Sμ(x) := ∑mn≤ x μ((m,n)). In this paper, we present an asymptotic formula for Sμ(x). Assuming the Riemann Hypothesis, we delve further into the asymptotic behavior of Sμ(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.