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.
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.
The paper deals with numerical computation of the asymptotic variance of the so-called increment ratio (IR) statistic and its modifications. The IR statistic is useful for estimation and hypothesis testing on fractional parameter H ∈ (0, 1) of random process (time series), see Surgailis et al. [1], Bardet and Surgailis [2]. The asymptotic variance of the IR statistic is given by an infinite integral (or infinite series) of 4-dimensional Gaussian integrals which depend on parameter H. Our method can be useful for numerical computation of other similar slowly convergent Gaussian integrals/series. Graphs and tables of approximate values of the variances σp2(H) and σˆp2(H), p = 1, 2 are included.
In the paper, an universality theorem of discrete type on the approximation of analytic functions by shifts of a special absolutely convergent Dirichlet series is obtained. These series is close in a certain sense to the periodic zeta-function and depends on a parameter.
In the paper, we construct an absolutely convergent Dirichlet series which in the mean is close to the periodic Hurwitz zeta-function, and has the universality property on the approximation of a wide class of analytic functions.
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