Complex Systems, Numbers and Number Theory

Abstract

All is number. PythagorasMuch ink has dried in the historical discussion of what is mathematics? Does it follows the scientific method? Is it just a collection of recipes, formulas and algorithms? Is it a science or a tool for others to use? Is it both? Carl Friedrich Gauss, one of the greatest mathematicians of all times called, it The Queen of Sciences but it was Eric Temple Bell who named it as Queen and Servant of Science in the title of one of his most celebrated books [1]. To some, this is an adequate way of expressing this dichotomy on the nature of mathematics and of its practical role ahead of other sciences [2]. There is, however, an aspect often neglected in all this discussion which is related to mathematics seen as natural phenomena and so susceptible of study by other sciences, such as physics. Do numbers along with their arithmetic operations behave as physical systems? More specifically, can the modern theory of nonlinear physics address number arithmetics as complex physical phenomena? From the remarkable coincidence in 1972 between H. Montgomery's work on the statistics of the spacings between zeta zeros and F. Dyson's analogous work on eigenvalues of random matrices, we have seen, somewhat unexpectedly, how number theory and physics have tended bridges between each other. These connections range from the reinterpretation of the Riemann zeta function as a partition function [3] or the focus of the Riemann Hypothesis via quantum chaos [4][5][6], to multifractality in the distribution of primes [7] or computational phase transitions in the number partitioning problem [8], to cite but a few (see [9] for an extensive bibliography).