Joint Discrete Approximation of a Pair of Analytic Functions by Periodic Zeta-Functions

Abstract

In the paper, the problem of simultaneous approximation of a pair of analytic functions by a pair of discrete shifts of the periodic and periodic Hurwitz zeta-function is considered. The above shifts are defined by using the sequence of imaginary parts of non-trivial zeros of the Riemann zeta-function. For the proof of approximation theorems, a weak form of the Montgomery pair correlation conjecture is applied.

Universality of Zeta-Functions of Cusp Forms and Non-Trivial Zeros of the Riemann Zeta-Function

“…The function ζ(s) has infinitely many so-called non-trivial zeros = β + iγ lying in the strip {s ∈ C : 0 < σ < 1}. By the Riemann hypothesis, all non-trivial zeros of ζ(s) lie on the critical line σ = 1 2 . Thus, let 0 < γ 1 < γ 2 < ... γ k ... be the sequence of imaginary parts of non-trivial zeros of the function ζ(s).…”

“…In [11], the joint approximation by shifts of Dirichlet L-functions involving the sequence {γ k } was discussed. Finally, the paper [1] is devoted to a generalization of [7] for shifts of the periodic and periodic Hurwitz zeta-functions.…”

Universality in Short Intervals of the Riemann Zeta-Function Twisted by Non-Trivial Zeros

“…was required. Analogical results for more general functions were given in [14,15]. On the other hand, all above theorems are non-effective in the sense that any concrete shift approximating a given analytic function is not known.…”

Joint universality of periodic zeta-functions with multiplicative coefficients. II

“…and its meromorphic continuation are called the periodic Hurwitz zeta-function. In [15] and [3], under hypothesis (1), joint universality theorems involving sequence {γ k } for the pair consisting from the Riemann and Hurwitz zeta-functions and their periodic analogues, respectively, were obtained, while in [23], such theorems were proved for Hurwitz zeta-functions.…”