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

Abstract

In this paper, I utilize a variant of the Selberg--Delange method to find quantitative estimates of the sums \[M_{k,ω}(x,y)=\sum_{\substack{p_{1}(n)> y\\ n\leq x} } μ(n) {ω(n)-1\choose k-1},\] where $y$ can grow with $x$ but we must have $y\le Y_0\exp (p\frac{\log x}{(\log \log (x+1){)}^{1+\epsilon }})$ with $Y_0,p,\epsilon >0$. 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 to 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.