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

In this paper, we analyze the nontrivial zeros of the Riemann zeta-function using the mul-tidimensional scaling (MDS) algorithm and computational visualization features. The non -trivial 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 em-ploy 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. (c) 2021 Elsevier B.V. All rights reserved.

Automated summary

In this paper, the nontrivial zeros of the Riemann zeta-function are analyzed using the multidimensional scaling (MDS) algorithm and computational visualization features. Different metrics produce different structures, with the Lorentzian metric revealing clear periodic patterns.