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

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 Möbius function $\hat{I}¼(n)$ 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 Möbius function and $\ddot{I}‰(n{)}^{k-1}$, where $\ddot{I}‰(n)$ is the number of distinct prime factors on $n$. In 2024, the first author and Jason Johnson proved new results involving $\hat{I}¼(n)$ and $\ddot{I}‰(n)$, 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 $k\ge 2$, $$\sum _{n=2}^{\infty }\frac{\hat{I}¼(n)\ddot{I}‰(n{)}^k}{n}=0,$$ where $\hat{I}¼(n)$ is the Möbius Function and $\ddot{I}‰(n)$ 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 $1\le j\le \ell$ and $(j,\ell )=1$ $$\sum _{\begin{array}{c}n=2\\ p_1(n)\equiv j(mod\ell )\end{array}}^{\infty }\frac{\hat{I}¼(n)\ddot{I}‰(n{)}^{k-1}}{n}=0,$$ for every $k\ge 3$; 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 $1\le j\le \ell$ and $(j,\ell )=1$ $$(-1{)}^k\sum _{\begin{array}{c}n=2\\ p_1(n)\equiv j(mod\ell )\end{array}}^{\infty }\frac{\hat{I}¼(n)\left(\genfrac{}{}{0ex}{}{\ddot{I}‰(n)-1}{k-1}\right)}{n}=\frac{1}{\ddot{I}†(\ell )},$$ for every $k\ge 3$. All results are established here in quantitative form.