A Weighted Discrete Universality Theorem for Periodic Zeta-Functions. Ii

Abstract

In the paper, a weighted theorem on the approximation of a wide class of analytic functions by shifts ζ(s + ikαh; a), k ∈ N, 0 < α < 1, and h > 0, of the periodic zeta-function ζ(s; a) with multiplicative periodic sequence a, is obtained.

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

“…If the above quantities are zero, then the corresponding zetafunctions are entire. The approximation of analytic functions by the functions ζ(s; a) and ζ(s, α; b) was studied in [8,26,28,29] and [2,7,18,22,24,25,27], respectively. The first joint results for a pair of functions ζ(s; a), ζ(s, α; b) has been obtained in [9].…”

Weighted Discrete Universality of the Riemann Zeta-Function

“…A generalization of Theorem 3 for Matsumoto zeta-functions was given in [12]. In [17], a weighted discrete universality theorem with the sequence {k α h}, 0 < α < 1, for the periodic zeta-function was obtained.…”

Discrete Universality of Absolutely Convergent Dirichlet Series

“…The majority of the papers deal with the continuous universality of ζ(s; a) when τ in shifts ζ(s + iτ ; a) takes arbitrary real values. To our knowledge, the discrete universality of ζ(s; a), when τ in ζ(s+iτ ; a) takes real values from a certain discrete set, was discussed only in [3,4,5] with multiplicative sequence a, i. e., a mn = a m a n for all (m, n) = 1 and a 1 = 1. For example, Theorem 2 of [4] with w(u) ≡ 1 implies the following result.…”

“…To our knowledge, the discrete universality of ζ(s; a), when τ in ζ(s+iτ ; a) takes real values from a certain discrete set, was discussed only in [3,4,5] with multiplicative sequence a, i. e., a mn = a m a n for all (m, n) = 1 and a 1 = 1. For example, Theorem 2 of [4] with w(u) ≡ 1 implies the following result. Denote by K the class of compact subsets of the strip D with connected complements, and by H 0 (K) with K ∈ K the class of continuous non-vanishing functions on K that are analytic in the interior of K. Let P be the set of all prime numbers, #A denotes the cardinality of the set A, and, for h > 0,…”