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

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.