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We introduce a novel arithmetic function $w(n)$, a generalization of theLiouville function $\lambda(n)$, as the coefficients of a Dirichlet series, andas a special case of a parametrized family of functions $w_m(n)$. We prove someuseful special properties of these arithmetic functions and then focus onconvergence of their Dirichlet series. In particular, we show that eachfunction $w_m(n)$ injectively maps $\mathbb{N}$ into a dense subset of the unitcircle in $\mathbb{C}$ and that $F_m(s) = \sum_n \frac{w_m(n)}{n^s}$ convergesfor all $s$ with $\Re(s)\in\left(\frac{1}{2},1\right)$. Finally, we show thatthe family of functions $w_m(n)$ converges to $\lambda(n)$ and that $F_m(s)$converges uniformly in $m$ to $\sum_n \frac{\lambda(n)}{n^s}$, implyingconvergence of that series in the same region and thereby proving aparticularly interesting property about a closely related function.

A Novel Generalization of the Liouville Function $\lambda(n)$ and a Convergence Result for the Associated Dirichlet Series