Duality Between Prime Factors and The Prime Number Theorem For Arithmetic Progressions -- Higher Order Dualities

TL;DR

This paper extends duality results involving prime factors, the Möbius function, and the Prime Number Theorem for Arithmetic Progressions to all higher orders, providing new summation identities and density theorems.

Contribution

It generalizes existing duality identities to all higher orders, linking prime factor distributions with advanced number theoretic functions and the Prime Number Theorem.

Findings

Proves that for each k≥2, the sum of μ(n)ω(n)^k/n over n converges to zero.

Establishes that for coprime j, ℓ, certain sums involving μ(n) and ω(n)^k-1 vanish for all k≥3.

Recasts results as density-type theorems relating prime residue classes and Möbius function sums.

Abstract

In 1977, the first author observed a duality between the largest and smallest prime factors of integers, and established as a consequence some new results on the M\"obius function $\mu(n)$ using the Prime Number Theorem for Arithmetic Progressions. In that 1977 paper, higher order dualities were observed involving the $k$-th largest and $k$-th smallest prime factors, facilitated by the M\"obius function and $\omega(n)^{k-1}$, where $\omega(n)$ is the number of distinct prime factors on $n$. In 2024, the first author and Jason Johnson proved new results involving $\mu(n)$ and $\omega(n)$, by exploiting the second order duality identity of Alladi (1977). We establish here extensions to all higher orders $k$, the results of Alladi (1977) and of Alladi-Johnson (2024), by utilizing the $k$-th order duality in Alladi's 1977 paper. First, we show that for each $k\geq 2$, $$ \sum_{n=2}^{\infty} \frac{\mu(n)\omega(n)^{k}}{n} =0, $$ where $\mu(n)$ is the M\"obius Function and $\omega(n)$ counts the number of distinct prime factors of $n$. Further, using the General Duality Identity and the Prime Number Theorem of Arithmetic Progressions, we prove that for integers $j,\ell$ satisfying $1 \leq j \leq \ell$ and $(j,\ell)=1$ $$ \sum_{\substack{n=2 \\ p_1(n) \equiv j\;(mod\;\ell)}}^{\infty} \frac{\mu(n)\omega(n)^{k-1}}{n}=0, \nonumber $$ for every $k \geq 3$; this result for $k=1$ is due to Alladi (1977) and for $k=2$ due to Alladi-Johnson (2024). We also recast this result in the following manner as a density-type theorem: for integers $j,\ell$ satisfying $1 \leq j \leq \ell$ and $(j,\ell)=1$ $$ (-1)^k\sum_{\substack{n=2 \\ p_1(n) \equiv j\;(mod\;\ell)}}^{\infty} \frac{\mu(n){\omega(n)-1 \choose k-1}}{n}=\frac{1}{\varphi(\ell)}, \nonumber $$ for every $k \geq 3$. All results are established here in quantitative form.