Universality of composite functions of periodic zeta functions

In the paper, we prove the universality, in the sense of Voronin, for some classes of composite functions $F(\zeta(s;\mathfrak a))$, where the function $\zeta(s;\mathfrak a)$ is defined by a Dirichlet series with periodic multiplicative coefficients. We also study the universality of functions of the form $F(\zeta(s;\mathfrak a_1),\dots,\zeta(s;\mathfrak a_r))$. For example, it follows from general theorems that every linear combination of derivatives of the function $\zeta(s;\mathfrak a)$ and every linear combination of the functions $\zeta(s;\mathfrak a_1),\dots,\zeta(s;\mathfrak a_r)$ are universal.
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