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

作     者：Machado, J. Tenreiro Luchko, Yuri 

作者机构：Polytech Porto Dept Elect Engn Inst Engn Porto Portugal Beuth Tech Univ Appl Sci Berlin Dept Math Phys & Chem Luxemburger Str 10 D-13353 Berlin Germany 

出 版 物：《COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION》 (非线性科学和数值模拟快报)

年 卷 期：2021年第102卷

页      面：105924-105924页

核心收录：

学科分类：07[理学] 0701[理学-数学] 0702[理学-物理学] 0801[工学-力学（可授工学、理学学位）] 070101[理学-基础数学] 

主　　题：Riemann's zeta-function Zeros of the zeta-function Distribution of zeros Complex systems Lorentzian metric Multidimensional scaling algorithm Periodical patterns 

摘      要：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.