Universality for $L$ -functions from the Selberg class

Universality of zeta and $L$-functions is one of the most interesting phenomenons of analytic number theory. Roughly speaking, it means that every analytic function can be approximated with a given accuracy by shifts of the considered zeta or $L$-functions, uniformly on compact subsets of a certain region.
In the report, we will focus on the universality of $L$-functions from Selberg's class [1], which is one of the most extensively studied objects in analytic number theory. We will present two types' results – continuous and discrete universality – when the shift can take arbitrary real values or values from a certain discrete set (e.g., from arithmetic progression), respectively. More precisely, the results given in [2], [3] and [4] will be discussed.
[1] A. Selberg, Old and new conjectures and results about a class of Dirichlet series. In: Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), E. Bombieri et al. (Eds.). Univ. Salerno. Salerno, 1992. P. 367 – 385.
[2] H. Nagoshi, J. Steuding, Universality for $L$-functions in the Selberg class. Lith. Math. J. 50:3 (2010). P. 393 – 411.
[3] R. Macaitienė, Mixed joint universality for $L$ -functions from Selberg's class and periodic Hurwitz zeta-functions. Chebysh. Sb., 16:1 (2015). P. 219 – 231.
[4] A. Laurinčikas, R. Macaitienė, Discrete universality in the Selberg class. Proceedings of the Steklov Institute of Mathematics. 2017. V. 299. (to appear).