Abstract
It is well known that the Riemann zeta-function is universal in the Voronin sense, i.e., its shifts ζ(s + iτ), τ ϵ R, approximate a wide class of analytic functions. The universality of ζ(s) is called discrete if τ take values from a certain discrete set. In the paper, we obtain a weighted discrete universality theorem for ζ(s) when τ takes values from the arithmetic progression {kh : k ϵN} with arbitrary fixed h > 0. For this, two types of h are considered.
Highlights
The Riemann zeta-function ζ(s), s = σ + it, ∞1 ζ(s) =ms, σ > 1, m=1 since Riemann’s and even Euler’s times surprises mathematicians by the extensive field of applications and denseness of the set of its values
1, are dense in a much more important merit of Voronin is his so-called universality theorem for the function ζ(s) [26]
We note that discrete universality theorems for zeta-functions sometimes are more convenient for practical applications, an example of this is the paper [3]
Summary
Ms , σ > 1, m=1 since Riemann’s and even Euler’s times surprises mathematicians by the extensive field of applications and denseness of the set of its values. A much more important merit of Voronin is his so-called universality theorem for the function ζ(s) [26]. By Theorem 1, the set of shifts ζ(s + iτ ) approximating a given function from H0(K) has a positive lower density, it is infinite. We note that discrete universality theorems for zeta-functions sometimes are more convenient for practical applications, an example of this is the paper [3]. In [10], Theorem 3 was proved under a certain additional hypothesis on the function w(t) which is a weighted version of the classical. A generalization of Theorem 3 for Matsumoto zeta-functions was given in [12]. The aim of this paper is a weighted discrete universality theorem for the Riemann zeta-function. For proving of the above universality theorems, we will apply the probabilistic approach
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
