| Abstract [eng] |
In 1989, A. Selberg defined some class S of Dirichlet series having an Euler product, analytic continuation and a functional equation of Riemann-type. Moreover, he formulated some fundamental conjectures concerning them. In the meantime, this so-called Selberg class became an important object of research, but still it is not understood very well. In the Master thesis, we investigate universality property of functions in the Selberg classS. J. Steuding has proved that functions inS that satisfy the mean value condition on primes are universal in the sense of Voronin, i.e., every function in a sufficiently wide classofanalyticfunctionscanbeapproximatedbytheshifts L(s+iτ), τ ∈R. Inthisthesis we show that every function in the same class of analytic functions can be approximated by the discrete shifts L(s+ikh), k =0,1,..., where h > 0 is an arbitrary fixed number. Let K be the class of compact subsets of D = {s ∈ C : _σL < σ < 1} (where σ_L = max{/2, 1− 1/d_L}) with connected complements and H_0(K), K ∈K, be the class of continuous functions that have no zeros in K and are analytic in the interior of K. Themaintheorem. Supposethatthe L-functionbelongstoS andsatisfiesadditional condition lim x→∞ 1/π(x)sum p≤x |a(p)|^2 = κ, κ > 0. Let K ∈K, f(s)∈ H_0(K). Then for every ε > 0 lim N→∞ inf 1/(N +1)#(0≤ k ≤ N :sup s∈K|L(s+ikh)−f(s)| < ε)> 0. To prove the main theorem we apply a probability method based on limit theorems on the weak convergence of probability measures in the space of analytic functions. Let P be the set of all primes and L(P,h,π)=(logp : p ∈P), π/h. We considered separately two cases: (1) the set L(P,h,π) is linearly independent over the field of rational numbers Q; (2) the set L(P,h,π) is linearly dependent over Q. Then the proof of the main theorem was based on limit theorems proved in 4.1 and 4.2 subsections as well as on the Mergelyan theorem on the approximation of analytic functions by polynomials. |
