A Discrete Limit Theorem for L-Functions of Elliptic Curves

Abstract

In the paper, we prove the discrete limit theorem in the sense of the weak convergence of probability measures in the space of analytic on D_V = {s ∈ C : 1 < σ < 3/2, |t| <  V} functions for L-functions of elliptic curves L_E(s). The main statement of the paper is as follows. Let h > 0 be a fixed real number such that exp {2πk/h} is an irrational number for all k∈Z\{0}. Then the probability measure μ_N(L_E(s + imh)∈A), A ∈ B(H(D_V)), converges weakly to the measure P_LE as N→∞.