Joint universality for dependent \(L\)-functions (Q1696812)

Let \(\chi_1,\dots,\chi_n\) be Dirichlet characters, \(\alpha_1,\dots,\alpha_n \in {\mathbb R}\), \(a_1,\dots,a_n>0\). Suppose \(b_j \in {\mathbb R}\) if \(a_j \notin {\mathbb Z}\) and \(b_j \in (-\infty,0] \cup (1,+\infty)\) if \(a_j \in {\mathbb N}\). Let also \(a_j \ne a_k\) or \(b_j \ne b_k\) if \(j \ne k\). The author proves that then suitable shifts of the type \(L(s+i\alpha_j t^{a_j} \log^{b_j} t;\chi_j)\) can simultaneously approximate any given set of analytic functions on a simply connected compact subset of the right open half of the critical strip. A discrete version of this statement (when \(t\) runs over the set \({\mathbb N}\)) is also obtained.