This talk concerns the non-vanishing and cyclicity of Dirichlet series of the form $D_{fg}(s)=\sum_{n=1}^{\infty}f(n)g(n)n^{-s}$, where $f$ is periodic and $g$ is completely multiplicative. Under the assumption that the sequence $\{g(p_j)\}_{j\geq1}$ is standard—satisfying conditions inspired by the sequence of reciprocals of primes $\{\frac{1}{2},\frac{1}{3},\frac{1}{5},\ldots,\frac{1}{p_j},\ldots\}$—we prove that the following are equivalent: (1) $D_{fg}$ is non-vanishing in the right half-plane; (2) $D_{fg}$ is cylclic in the Hardy space $\mathcal{H}^2$ of Dirichlet series; (3) $D_{fg}$ can be factorized as a product of a non-vanishing Dirichlet polynomial and a Dirichlet series with completely multiplicative coefficients.
