Joint value distribution theorems for the Riemann and Hurwitz zeta-functions

In the paper, a class of functions $\varphi(t)$ is introduced such that a given pair of analytic functions is approximated simultaneously by shifts $\zeta(s+i\varphi(k)),\zeta(s+i\varphi(k),\alpha)$, $k\in\mathbb N$, of the Riemann and Hurwitz zeta-functions with parameter $\alpha$ for which the set $\{(\log p\colon p\ \text{is prime}),\ (\log(m+\alpha)\colon m\in\mathbb N_0)\}$ is linearly independent over $\mathbb Q$. The definition of this class includes an estimate for $\varphi(t)$ and $\varphi'(t)$ as well as uniform distribution modulo 1 of the sequence $\{a\varphi(k)\colon k\in\mathbb N\}$, $a\neq0$.