Joint Discrete Universality in the Selberg–Steuding Class

Abstract

In the paper, we consider the approximation of analytic functions by shifts from the wide class S˜ of L-functions. This class was introduced by A. Selberg, supplemented by J. Steuding, and is defined axiomatically. We prove the so-called joint discrete universality theorem for the function L(s)∈S˜. Using the linear independence over Q of the multiset (hjlogp:p∈P),j=1,…,r;2π for positive hj, we obtain that there are many infinite shifts L(s+ikh1),…,L(s+ikhr), k=0,1,…, approximating every collection f1(s),…,fr(s) of analytic non-vanishing functions defined in the strip {s∈C:σLσ1}, where σL is a degree of the function L(s). For the proof, the probabilistic approach based on weak convergence of probability measures in the space of analytic functions is applied.

Keywords

• analytic functions
• discrete shifts
• limit theorem
• simultaneous approximation
• Selberg–Steuding class
• weak convergence