The Investigation of the Discrete Universality of L-Functions of Elliptic Curves

Abstract

In the paper, we prove the discrete universality theorem in the sense of the weak convergence of probability measures in the space of analytic functions for the L-functions of elliptic curves. We consider an approximation of analytic functions by translations LE (s+imh) , where h > 0 is a fixed number, the translations of the imaginary part of the complex variable take values from some discrete set such as arithmetical progression. We suppose that the number h > 0 is chosen so that exp{2πk/h } is an rational number for some non-zero integer. The proof of discrete universality of the derivatives of L-functions of elliptic curves is based on a limit theorem in the sense of weak convergence of probability measures in the space of analytic functions.