Zeros of the Riemann zeta-function and its universality

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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.

Section: Introductionmentioning

confidence: 99%

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

Balčiūnas

Garbaliauskienė

Karaliūnaitė

et al. 2020

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“…We recall that the condition (1.1) for the first-time was applied in [5] for the approximation by shifts ζ(s + iγ k h), and in [7] for joint approximation by…”

mentioning

confidence: 99%

“…For the proof of Theorem 3, we will apply some results from [5] and [8]. On the mentioned results, we will construct a probabilistic model.…”

mentioning

confidence: 99%

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

Balčiūnas

Franckevič

Garbaliauskienė

et al. 2021

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It is known that zeta-functions ζ(s,F) of normalized Hecke-eigen cusp forms F are universal in the Voronin sense, i.e., their shifts ζ(s + iτ,F), τ R, approximate a wide class of analytic functions. In the paper, under a weak form of the Montgomery pair correlation conjecture, it is proved that the shifts ζ(s+iγkh,F), where γ1 < γ2 < ... is a sequence of imaginary parts of non-trivial zeros of the Riemann zeta function and h > 0, also approximate a wide class of analytic functions.

“…For shifts ζ(s + ik α h) with fixed α, 0 < α < 1, Theorem 2 is given in [6]. In [15,21] and [7,8,13,16], more general shifts of Dirichlet L-functions and Riemann zeta-function, respectively, were considered. We note that discrete universality theorems for zeta-functions sometimes are more convenient for practical applications, an example of this is the paper [3].…”

Section: Introductionmentioning

confidence: 99%

Weighted Discrete Universality of the Riemann Zeta-Function

Laurinčikas

Šiaučiūnas

Vadeikis

2020

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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.