Multidimensional scaling and visualization of patterns in distribution of nontrivial zeros of the zeta-function

•Pattern visualization of the zeta-function zeros.•Multidimensional scaling analysis using Lorentzian metric.•Periodical patterns in the scope of Riemann hypothesis. In this paper, we analyze the nontrivial zeros of the Riemann zeta-function using the multidimensional scaling (MDS) algorithm and com...

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Vydáno v:Communications in nonlinear science & numerical simulation Ročník 102; s. 105924
Hlavní autoři: Machado, J. Tenreiro, Luchko, Yuri
Médium: Journal Article
Jazyk:angličtina
Vydáno: Amsterdam Elsevier B.V 01.11.2021
Elsevier Science Ltd
Témata:
Algorithms
Complex systems
Distribution of zeros
Lorentzian metric
Mathematical analysis
Multidimensional scaling algorithm
Number theory
Periodical patterns
Riemann’s zeta-function
Visualization
Zeros of the zeta-function
Distribution of zeros
Zeros of the zeta-function
Riemann’s zeta-function
Multidimensional scaling algorithm
Periodical patterns
Complex systems
Lorentzian metric
ISSN:1007-5704, 1878-7274
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Shrnutí:•Pattern visualization of the zeta-function zeros.•Multidimensional scaling analysis using Lorentzian metric.•Periodical patterns in the scope of Riemann hypothesis. In this paper, we analyze the nontrivial zeros of the Riemann zeta-function using the multidimensional scaling (MDS) algorithm and computational visualization features. The nontrivial zeros of the Riemann zeta-function as well as the vectors with several neighboring zeros are interpreted as the basic elements (points or objects) of a data set. Then we employ a variety of different metrics, such as the Jeffreys and Lorentzian ones, to calculate the distances between the objects. The set of the calculated distances is then processed by the MDS algorithm that produces the loci, organized according to the objects features. Then they are analyzed from the perspective of the emerging patterns. Surprisingly, in the case of the Lorentzian metric, this procedure leads to the very clear periodical structures both in the case of the objects in form of the single nontrivial zeros of the Riemann zeta-function and in the case of the vectors with a given number of neighboring zeros. The other tested metrics do not produce such periodical structures, but rather chaotic ones. In this paper, we restrict ourselves to numerical experiments and the visualization of the produced results. An analytical explanation of the obtained periodical structures is an open problem worth for investigation by the experts in the analytical number theory.
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ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2021.105924