On subradically sifted sums related to Alladi's higher order duality between prime factors

In this paper, I utilize a variant of the Selberg–Delange method to find quantitative estimates of the sums $$M_{k,\omega }(x,y)=\sum _{\begin{array}{c}{p}_1(n)>y\\ n\le x\end{array}}\mu (n)\left(\genfrac{}{}{0px}{}{\omega (n)-1}{k-1}\right),$$ where $y$ can grow with $x$ but we must have $y\le Y_0\exp (\mathcal{p}\frac{\log x}{(\log \log (x+1){)}^{1+\epsilon }})$ with $Y_0,\mathcal{p},\epsilon >0$. Moreover, I give preliminary upper bounds for the general range $1.9\le y\le x^{\frac{1}{k}}$. In addition, I formalize the notions of subradical and radical dominance and discuss their relevance to the analytic approach of the study of arithmetic functions. Lastly, I give a fascinating formula related to the derivatives of the gamma function and the Hankel contour, which should be relevant for those employing the Selberg–Delange method to obtain higher-order terms.