On joint value distribution of Hurwitz zeta-functions

It is well known that some zeta and $L$-functions are universal in the Voronin sense, i.e., they approximate a wide class of analytic functions. Also, some of them are jointly universal. In this case, a collection of analytic functions is simultaneously approximated by a collection of zeta-functions. In the paper, a problem related to joint universality of Hurwitz zeta-functions is discussed. It is known that the Hurwitz zeta-functions $\zeta(s,\alpha_1), \dots, \zeta(s,\alpha_r)$ are jointly universal if the parameters $\alpha_1,\dots, \alpha_r$ are algebraically independent over the field of rational numbers $\mathbb{Q}$, or, more generally, if the set $\{\log(m+\alpha_j): m\in \mathbb{N}_0,\; j=1,\dots, r\}$ is linearly independent over $\mathbb{Q}$. We consider the case of arbitrary parameters $\alpha_1,\dots, \alpha_r$ and obtain that there exists a non-empty closed set $F_{\alpha_1,\dots, \alpha_r}$ of the space $H^r(D)$ of analytic functions on the strip $D=\left\{s\in \mathbb{C}:\frac{1}{2}<\sigma<1\right\}$ such that, for every compact sets $K_1,\dots, K_r\subset D$, $f_1,\dots, f_r\in F_{\alpha_1,\dots, \alpha_r}$ and $\varepsilon>0$, the set $\left\{\tau\in \mathbb{R}: \sup_{1\leqslant j\leqslant r} \sup_{s\in K_j} |\zeta(s+i\tau,\alpha_j)-f_j(s)|<\varepsilon\right\}$ has a positive lower density. Also, the case of positive density of the latter set is discussed.