A Weighted Universality Theorem for Periodic Zeta-Functions

Abstract

The periodic zeta-function ζ(s; a), s = σ + it is defined for σ > 1 by the Dirichlet series with periodic coefficients and is meromorphically continued to the whole complex plane. It is known that the function ζ(s; a), for some sequences a of coefficients, is universal in the sense that its shifts ζ(s + iτ ; a), τ ∈ R, approximate a wide class of analytic functions. In the paper, a weighted universality theorem for the function ζ(s; a) is obtained.

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

“…The next lemma deals with the approximation of ζ(s; a) by ζ n (s; a). Denote by ρ the metric in H(D), see, for example, [18]. holds.…”

“…Generalizations of a theorem of such a type were given in [9] and [4]. The weighted universality for the function ζ(s; a) was began to study in [18]. We remind the main result of [18].…”

“…The first universality results for ζ(s; a) were obtained in [1], [2], [21] and [22]. The universality of ζ(s; a) with multiplicative sequence a was considered in [16], [23], [18] and [17]. We remind the paper [6], where a new type of universality for the function ζ(s; a) was introduced.…”

“…The weighted universality for the function ζ(s; a) was began to study in [18]. We remind the main result of [18]. Letŵ(t) be a positive function of bounded variation on…”

“…Let K be the class of compact subsets of the strip D = s ∈ C : 1 2 < σ < 1 with connected complements, and let H 0 (K), K ∈ K, be the class of continuous non-vanishing functions on K which are analytic in the interior of K. Moreover, let I A denote the indicator function of the set A. We remind that the sequence a = {a m } is called multiplicative if a mn = a m a n for all coprimes m, n ∈ N. Now we state an universality theorem from [18]. Theorem 1.…”