Joint Universality of Certain Dirichlet Series

In this paper, we define the Dirichlet series $ \zeta_{u_T j} (s)$, $ j = 1, \dots, r$, absolutely converging in the half-plane $ \operatorname{Re} s> 1/2 $ and prove that the set of shifts $ (\zeta_{u_T 1} (s + ia_1 \tau), \dots, \zeta_{u_T r} (s + ia_r \tau)) $ approximating a given set of analytic functions has a positive density on the interval $ [T, T + H]$, $ H = o (T) $ as $ T \to \infty$. Here $ a_1, \dots, a_r \in \mathbb{R} $ are algebraic numbers linearly independent over $ \mathbb{Q} $ and $ u_T \to \infty $ as $ T \to \infty$.