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 \(1\le \ell \le k\) and \((\ell ,k)=1\) , \(\begin{aligned} \sum _{n\ge 2;\;p_1(n) \equiv \ell \;(mod\;k)}\frac{\mu (n)\omega (n)}{n} = 0, \end{aligned}\) where \(\mu (n)\) is the Möbius 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 \(\begin{aligned} \sum _{ n \ge 2;\;\left[ \frac{K/\mathbb {Q}}{p_1(n)}\right] =C} \frac{\mu (n)\omega (n)}{n} = 0. \end{aligned}\) 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.
