Approximation of Analytic Functions by Shifts of Certain Compositions

Šiaučiūnas, Darius; Šimėnas, Raivydas; Tekorė, Monika

doi:10.3390/math9202583

Cited by 3 publications

(2 citation statements)

References 27 publications

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“…Deeper consideration about analytic function can be done. For example, Ref [6] obtained theorems of multidimensional space of analytic function. The order of imaginary section of non-trivial zeros of the Riemann zeta-function is contained in the used shifts of periodic zeta-functions.…”

Section: Analytic Functionmentioning

confidence: 99%

The Application of Complex Variables Function and Residue Theorem

Hu¹,

Lan²,

Lin³

et al. 2023

HSET

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The history of complex function can be dated back to about 18 century when Euler came up with two equations derived from integrals of complex function. Over the age, mathematicians have been trying to explore and discuss more in this mysterious field, and this new branch gradually became prevalent in the 19 century mathematics like when calculus ruled the 18 century mathematics. In particular, the concept of residue was invented and became an important part in complex function, being applied in some special types of real integral. Also it is a generalization of Cauchy integral theorem and Cauchy integral formula. Nowadays, the area of complex function is filled with crystallization of the wisdom of countless scholars and researchers, and the precious mathematical treasure can also meet the needs of other academic areas like physics and biology. Summarizing the various approaches and examples of application in fields of mathematics and others can be a valuable topic. In this paper, the history of application and some examples for references of complex functions, and introduce the main concepts and formulas about them will be briefly discussed, including the area of complex numbers, analytic function, and residue. There are some relationships between some of the applications to be dig out and considerable real-world problems, and they are supposed to the generalization of those subjects can provide certain enlightenment to people interested in or during the study in fields of complex function.

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Section: Analytic Functionmentioning

confidence: 99%

The Application of Complex Variables Function and Residue Theorem

Hu¹,

Lan²,

Lin³

et al. 2023

HSET

View full text Add to dashboard Cite

show abstract

“…The discrete generalized shifts were used in universality theorems for zeta-and Lfunctions. For example, in a series of works, the sequence {γ k : k ∈ N} of imaginary parts of nontrivial zeros of the Riemann zeta-function in generalized shifts was used; see [18,19] and references therein. For the proof of universality, a limit theorem with generalized shifts ζ(s + ikγ k ; a) for periodic zeta-functions in the space of analytic functions was obtained in [18].…”

Section: Introductionmentioning

confidence: 99%

A Generalized Discrete Bohr–Jessen-Type Theorem for the Epstein Zeta-Function

Laurinčikas

Macaitienė

2023

Mathematics

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Suppose that Q is a positive defined n×n matrix, and Q[x̲]=x̲TQx̲ with x̲∈Zn. The Epstein zeta-function ζ(s;Q), s=σ+it, is defined, for σ>n2, by the series ζ(s;Q)=∑x̲∈Zn∖{0̲}(Q[x̲])−s, and it has a meromorphic continuation to the whole complex plane. Let n⩾4 be even, while φ(t) is an increasing differentiable function with a continuous monotonic bounded derivative φ′(t) such that φ(2t)(φ′(t))−1≪t, and the sequence {aφ(k)} is uniformly distributed modulo 1. In the paper, it is obtained that 1N#N⩽k⩽2N:ζ(σ+iφ(k);Q)∈A, A∈B(C), for σ>n−12, converges weakly to an explicitly given probability measure on (C,B(C)) as N→∞.

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Joint Universality in Short Intervals with Generalized Shifts for the Riemann Zeta-Function

Laurinčikas

2022

Mathematics

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In the paper, the simultaneous approximation of a tuple of analytic functions in the strip {s=σ+it∈C:1/2<σ<1} by shifts (ζ(s+iφ1(τ)),…,ζ(s+iφr(τ))) of the Riemann zeta-function ζ(s) with a certain class of continuously differentiable increasing functions φ1,…,φr is considered. This class of functions φ1,…,φr is characterized by the growth of their derivatives. It is proved that the set of mentioned shifts in the interval [T,T+H] with H=o(T) has a positive lower density. The precise expression for H is described by the functions (φj(τ))1/3(logφj(τ))26/15 and derivatives φj′(τ). The density problem is also discussed. An example of the approximation by a composition F(ζ(s+iφ1(τ)),…,ζ(s+iφr(τ))) with a certain continuous operator F in the space of analytic functions is given.

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Approximation of Analytic Functions by Shifts of Certain Compositions

Cited by 3 publications

References 27 publications

The Application of Complex Variables Function and Residue Theorem

The Application of Complex Variables Function and Residue Theorem

A Generalized Discrete Bohr–Jessen-Type Theorem for the Epstein Zeta-Function

Joint Universality in Short Intervals with Generalized Shifts for the Riemann Zeta-Function

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