Algebraic analogues of results of Alladi-Johnson using the Chebotarev Density Theorem

Abstract

\textit{{\small We aim to get an algebraic generalization of Alladi-Johnson's (A-J) work on Duality between Prime Factors and the Prime Number Theorem for Arithmetic Progressions - II, using the Chebotarev Density Theorem (CDT). It has been proved by A-J, that for all positive integers $k,\ell$ such that $1\leq \ell\leq k$ and $(\ell,k)=1$,}} \begin{equation} \sum_{n\geq 2;\;p_1(n) \equiv \ell\;(mod\;k)}\frac{\mu(n)\omega(n)}{n} = 0, \nonumber \end{equation} \textit{{\small where $\mu(n)$ is the M\"obius function, $\omega(n)$ is the number of distinct prime factors of $n$, and $p_1(n)$ is the smallest prime factor of $n$. In our work here, we will prove the following result: If $C$ is a conjugacy class of the Galois group of some finite extension $K$ of $\mathbb{Q}$, then}} \begin{equation} \sum_{ n \geq 2;\;\left[\frac{K/\mathbb{Q}}{p_1(n)}\right]=C} \frac{\mu(n)\omega(n)}{n} = 0. \nonumber \end{equation} \textit{{\small where $\left[\frac{K/\mathbb{Q}}{p_1(n)}\right]$ is the Artin symbol. When $K$ is a cyclotomic extension of $\mathbb{Q}$, this reduces to the exact case of A-J's result.}}