Joint approximation by Dirichlet $L$-functions
Keywords:Dirichlet L-functions, joint universality, functional independence, weak convergence
In the paper, collections of analytic functions are simultaneously approximated by collections of shifts of Dirichlet $L$-functions $\left(L(s+i\gamma_1(\tau), \chi_1),\right.$ $\left. \dots, L(s+i\gamma_r(\tau), \chi_r)\right)$, with arbitrary Dirichlet characters $\chi_1,\dots, \chi_r$. The differentiable functions $\gamma_1(\tau), \dots, \gamma_r(\tau)$ and their derivatives satisfy certain growth conditions. The obtained results extend those of .