A MIXED JOINT UNIVERSALITY THEOREM FOR ZETA-FUNCTIONS. II

Vaida Pocevičienė,Darius Šiaučiūnas

doi:10.3846/13926292.2014.893261

Abstract

In the paper, a joint universality theorem on the approximation of analytic functions for zeta-function of a normalized Hecke eigen cusp form and a collection of periodic Hurwitz zeta-functions with algebraically independent parameters is obtained.

Highlights

In the paper, a joint universality theorem on the approximation of analytic functions for zeta-function of a normalized Hecke eigen cusp form and a collection of periodic Hurwitz zeta-functions with algebraically independent parameters is obtained
A natural question arises if the Riemann zeta-function in Theorem 1 can be replaced by other zeta-functions which are universal in a certain strip?
For a region G on the complex plane, let us denote by H(G) the space of analytic functions on G equipped with the topology of uniform convergence on compacta

Summary

Functional Limit Theorems

For a region G on the complex plane, let us denote by H(G) the space of analytic functions on G equipped with the topology of uniform convergence on compacta. This, the continuity of the function hn and Theorem 5.1 from [1] together with Lemma 1 show that the measure PT,n converges weakly to Pn = mH h−n 1 as T → ∞. By the definition of the random element Xn(s, s), Pn Hεv ≥ 1 − ε for all n ∈ N This means that the family of probability measures {Pn : n ∈ N} is tight, and, by the Prokhorov theorem, it is relatively compact. Using an equivalent of the weak convergence of probability measures in terms of continuity sets, Theorem 2.1 of [1], we have by Lemma 5 that lim νT ζ(s + iτ, s + iτ, α, ω; a, F ) ∈ A = P (A).

The Support of the Measure Pζ

Proof of Theorem 3