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

TL;DR

This paper develops estimates for sums involving the Möbius function and prime factors using a variant of the Selberg--Delange method, introduces concepts of subradical and radical dominance, and explores higher-order terms via gamma function derivatives.

Contribution

It introduces a new approach to estimate sums over prime factors with a growth condition on y, formalizes subradical and radical dominance, and connects gamma function derivatives to the Selberg--Delange method.

Findings

Derived quantitative estimates for sums involving the Möbius function and prime factors.

Established bounds for the sums in a specific range of y relative to x.

Presented a novel formula involving gamma function derivatives and the Hankel contour.

Abstract

In this paper, I utilize a variant of the Selberg--Delange method to find quantitative estimates of the sums \[M_{k,\omega}(x,y)=\sum_{\substack{p_{1}(n)> y\\ n\leq x} } \mu(n) {\omega(n)-1\choose k-1},\] where $y$ can grow with $x$ but we must have $y\leq Y_0\exp(\mathscr{p}\frac{\log x}{(\log\log (x+1))^{1+\epsilon}})$ with $Y_0,\mathscr{p},\epsilon>0$. Moreover, I give preliminary upper bounds for the general range $1.9\leq y\leq x^{\frac{1}{k}}$. 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.