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

TL;DR

This paper applies multidimensional scaling and visualization techniques to analyze the distribution of nontrivial zeros of the Riemann zeta-function, revealing periodic structures with Lorentzian metrics.

Contribution

It introduces a novel application of MDS with different metrics to visualize and identify patterns in the zeros of the zeta-function, highlighting Lorentzian metric's unique results.

Findings

Lorentzian metric reveals clear periodic structures

Other metrics produce chaotic patterns

Visualization aids in pattern recognition in zeta zeros

Abstract

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 Euclidean 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.