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

$\mathit{\text{We}}\mathit{\text{aim}}\mathit{\text{to}}\mathit{\text{get}}\mathit{\text{an}}\mathit{\text{algebraic}}\mathit{\text{generalization}}\mathit{\text{of}}\mathit{\text{Alladi-Johnson’s}}\mathit{\text{(A-J)}}\mathit{\text{work}}\mathit{\text{on}}\mathit{\text{Duality}}\mathit{\text{between}}\mathit{\text{Prime}}\mathit{\text{Factors}}\mathit{\text{and}}\mathit{\text{the}}\mathit{\text{Prime}}\mathit{\text{Number}}\mathit{\text{Theorem}}\mathit{\text{for}}\mathit{\text{Arithmetic}}\mathit{\text{Progressions}}\mathit{\text{-}}\mathit{\text{II,}}\mathit{\text{using}}\mathit{\text{the}}\mathit{\text{Chebotarev}}\mathit{\text{Density}}\mathit{\text{Theorem}}\mathit{\text{(CDT).}}\mathit{\text{It}}\mathit{\text{has}}\mathit{\text{been}}\mathit{\text{proved}}\mathit{\text{by}}\mathit{\text{A-J,}}\mathit{\text{that}}\mathit{\text{for}}\mathit{\text{all}}\mathit{\text{positive}}\mathit{\text{integers}}k,\mathrm{\ell }\mathit{\text{such}}\mathit{\text{that}}1\le \mathrm{\ell }\le k\mathit{\text{and}}(\mathrm{\ell },k)=1\mathit{\text{,}}$ \begin{equation} \sum_{n\geq 2;;p_1(n) \equiv \ell;(mod;k)}\frac{μ(n)ω(n)}{n} = 0, \nonumber \end{equation} $\mathit{\text{where}}\mu (n)\mathit{\text{is}}\mathit{\text{the}}\mathit{\text{M}}\ddot{\mathit{\text{o}}}\mathit{\text{bius}}\mathit{\text{function,}}\omega (n)\mathit{\text{is}}\mathit{\text{the}}\mathit{\text{number}}\mathit{\text{of}}\mathit{\text{distinct}}\mathit{\text{prime}}\mathit{\text{factors}}\mathit{\text{of}}n\mathit{\text{,}}\mathit{\text{and}}{p}_1(n)\mathit{\text{is}}\mathit{\text{the}}\mathit{\text{smallest}}\mathit{\text{prime}}\mathit{\text{factor}}\mathit{\text{of}}n\mathit{\text{.}}\mathit{\text{In}}\mathit{\text{our}}\mathit{\text{work}}\mathit{\text{here,}}\mathit{\text{we}}\mathit{\text{will}}\mathit{\text{prove}}\mathit{\text{the}}\mathit{\text{following}}\mathit{\text{result:}}\mathit{\text{If}}C\mathit{\text{is}}\mathit{\text{a}}\mathit{\text{conjugacy}}\mathit{\text{class}}\mathit{\text{of}}\mathit{\text{the}}\mathit{\text{Galois}}\mathit{\text{group}}\mathit{\text{of}}\mathit{\text{some}}\mathit{\text{finite}}\mathit{\text{extension}}K\mathit{\text{of}}\mathbb{Q}\mathit{\text{,}}\mathit{\text{then}}$ \begin{equation} \sum_{ n \geq 2;;\left[\frac{K/$\mathbb{Q}$}{p_1(n)}\right]=C} \frac{μ(n)ω(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.}}