Following in the footsteps of Selberg: an analogue of the Riemann hypothesis, a density theorem and a disrtibution law for the values of $L$-functions (and their linear combinations) on the critical line

In 1989, Atle Selberg introduced a class of $L$-series which are supposed to satisfy an analogue of the Riemann hypothesis. In our talk we shall speak about the following results proven for certain functions from the Selberg class:

• a positive proportion of non-trivial zeros of $L$-function lie on the critical line;
• almost all non-trivial zeros of $L$-function lie in a vicinity of the critical line;
• logarithm of $L$-function on the critical line is asymptotically normally distributed.

It turns out that in a certain sense all these results are equivalent and their core is a solution of the additive problem with the coefficients of the corresponding Dirichlet $L$-series.
We shall also talk about zeros of linear combinations of $L$-functions from the Selberg class and about a distribution of the values of such linear combinations on the critical line.