A new study led by Giuseppe Mussardo of SISSA uses statistical physics to shed new light on the mathematical problem of the millennium.
We may think we know the mathematical laws that govern the world, yet there are still many open problems to which scientists have not yet found a solution.
Between paradoxes, theorems, and conjectures, there is one problem in particular that is unanimously called the most important open problem in mathematics, which has engaged scientists around the world for over 150 years.
The Riemann conjecture
The problem in question was formulated by Bernhard Riemann in 1859, and is a complex hypothesis concerning the distribution of zeros on the so-called “zeta function”, the function of a complex variable – called precisely Zeta – which was already known in Euler’s time but is now referred to simply as Riemann’s zeta function.
The demonstration of the Riemann hypothesis would have important consequences on the distribution of prime numbers, the “holy grail” of the mathematical community.
Since the eighteenth century the most illustrious mathematicians have been searching for a mathematical rule able to demonstrate a logic in the distribution of prime numbers.
To this day, the distribution of prime numbers is considered random, and it is because of this that the prime numbers are the basis of encryption algorithms used in computer security.
But a demonstration of the Riemann hypothesis would have fundamental consequences also for quantum physics, for some years closely linked to attempts to demonstrate Riemann’s zeta function.
As early as 2017, a research group from the United States, Canada and the United Kingdom attempted to prove the conjecture using a mathematical tool peculiar to quantum mechanics (the Hamiltonian operator), and hypothesized that there might be a relationship between the zeros of the zeta function and the energy states of a quantum system.
And it is still from physics that more pieces arrive for the solution of the mathematical problem that has been open for over 160 years now.
Explaining mathematics with physics
The Riemann problem is among the seven mathematical problems of the millennium, the seven unsolved problems for the solution of which the famous Clay Institute in Cambridge has put up a million dollars.
The last piece that brings us closer to the possible solution of the mathematical problem of the millennium comes from the Journal of Statistical Mechanics, and bears the signature of Giuseppe Mussardo, of SISSA – Scuola Internazionale Superiore di Studi Avanzati, and André Leclair of Cornell University.
The two researchers have analyzed in the last three years an incredible amount of data, following a new approach based on statistical physics, in particular on the physics of chaotic motions.
The solution to the mathematical enigma opened for over a century could really come from physics, which could provide new important keys to the interpretation of the Riemann function.
The fact that it is physics, which has always been the handmaiden of mathematics, to provide tools and methods of interpretation for the resolution of a purely mathematical problem is exceptional, and the evolution of attempts to prove the Riemann hypothesis seems to show just this tendency.
The mission to understand the logic behind the arrangement of the zeros along the Riemann function is enriched, thanks to this study, by a possible explanation rather unexpected: in the straight-line distribution of the zeros there would be chaotic motion, together with the statistical laws that govern it. A new light on the mathematical conundrum of the century.
As the authors point out in the research, we are not faced with a rigorous proof of the Riemann hypothesis, but “if this work has the effect of stimulating further rigorous studies by real mathematicians on the subject, it will have already achieved its purpose of drawing attention to a possible way of dealing with a long-standing problem such as the Riemann hypothesis.”