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

Yazan Alamoudi

TL;DR

This work tackles quantitative estimates for the higher-order Alladi dualities by analyzing the sums $M_{k,\omega}(x,y)$ in a refined sifting window $y \le Y_0\exp\big(\mathscr{p}\frac{\log x}{(\log\log (x+1))^{1+\varepsilon}}\big)$ using a variant of the Selberg–Delange method. The author develops an explicit contour-integral framework with a Hankel-type path and introduces the constants $\Gamma_{m,N}$ to capture higher-order gamma-derivative contributions, yielding a main term of the form $\frac{x}{\log x}$ times a structured sum with $\Gamma_{i,j}$ and related coefficients, plus rigorous error bounds. A companion general upper-bound result extends the analysis to larger sifting ranges, giving subradical- and radical-growth-based interpretations via a new partial-order framework that clarifies the optimality of the chosen sifting rate. Together, these results advance the quantitative understanding of Alladi's higher-order dualities and provide tools for accessing higher-order terms in related analytic-arithmetic problems, with potential implications for the study of arithmetic functions under radical-like dominance.

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 $y\leq Y_0\exp(\mathscr{p}\frac{\log x}{(\log\log (x+1))^{1+ε}})$ with $Y_0,\mathscr{p},ε>0$. 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.

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

TL;DR

This work tackles quantitative estimates for the higher-order Alladi dualities by analyzing the sums in a refined sifting window using a variant of the Selberg–Delange method. The author develops an explicit contour-integral framework with a Hankel-type path and introduces the constants to capture higher-order gamma-derivative contributions, yielding a main term of the form times a structured sum with and related coefficients, plus rigorous error bounds. A companion general upper-bound result extends the analysis to larger sifting ranges, giving subradical- and radical-growth-based interpretations via a new partial-order framework that clarifies the optimality of the chosen sifting rate. Together, these results advance the quantitative understanding of Alladi's higher-order dualities and provide tools for accessing higher-order terms in related analytic-arithmetic problems, with potential implications for the study of arithmetic functions under radical-like dominance.

Abstract

In this paper, I utilize a variant of the Selberg--Delange method to find quantitative estimates of the sums where can grow with but we must have with . Moreover, I give preliminary upper bounds for the general range . 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.
Paper Structure (6 sections, 19 theorems, 176 equations)

This paper contains 6 sections, 19 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. An overview of the approach
  3. Controlling the error related to the sifting variable
  4. The quantitative statement for $y\leq Y_0\exp(\mathscr{p}\frac{\log x}{(\log\log (x+1))^{1+\epsilon}})$
  5. A general upper bound
  6. A general discussion on the subradical and radical strict partial orders

Key Result

Theorem 1.1

Let $N,k-1\in \mathbb{N}$ and $Y_0,\mathscr{p},\epsilon\in\mathbb{R}_{>0}$. If $y>x^{\frac{1}{k}}$, then $M_{k,\omega}(x,y)=0$. Otherwise, there is a constant $C_{N,k-1}$, that may depend on $N$, $k-1$, $Y_0$, $\epsilon$ and $\mathscr{p}$ but is otherwise absolute (in particular, $C_{N,k-1}$ does no and with and where $\phi_{i,j}(y)$ are some functions of $y$ given an explicit formula, in terms

Theorems & Definitions (43)

  • Theorem 1.1
  • Remark
  • Remark
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark
  • Theorem 3.1
  • Lemma 3.2
  • ...and 33 more