Primes, the Riemann Hypothesis, and the weighing of evidence

Number Theory is arguably one of the most fascinating subjects in Mathematics, while Theoretical Physics has helped us to shape our understanding of the laws of Nature. Both are characterized by the standards of clarity, beauty and depth. Sometimes the two subjects converge in a miraculous way, providing one of those vital, wonderful and superb narratives that are occasionally found in science.
Our story today concerns the Riemann Hypothesis, certainly the most famous open problem in mathematics but one that is not usually seen as being connected to physics. It states that the zeros of the Riemann zeta function (which is actually Euler’s!) lie on the critical line Re(s)=1/2, and similarly for other Dirichlet L-functions.  

The first talk, by Giuseppe Mussardo, will interpret and lend support to this hypothesis from the point of view of statistical physics, with Dirichlet L-functions regarded as quantum partition functions on the prime numbers and with the mechanism suggesting the truth of the Riemann Hypothesis being a kind of arithmetical analogue of Brownian motion. He will present the probabilistic arguments which lead to this conclusion and also discuss a battery of highly non-trivial tests which support the validity of this result with an extremely high confidence.

The second talk, by Don Zagier, will discuss both the Riemann Hypothesis and some other related problems of Number Theory from the more general point of view of the question of how to evaluate empirical evidence and when one can become convinced of the truth of an unproved mathematical assertion -- including a discussion of several well-known examples of statements for which the experimental evidence seemed to be overwhelmingly convincing but which nevertheless turned out to be false. If time permits, he would also give some examples from his own research of results reinterpreting the Riemann Hypothesis in unexpected ways.

Program

Prelude: What is the Riemann Hypothesis?

• Giuseppe Mussardo (Theoretical Physics Professor at the International School for Advanced Studies in Trieste)
• Don Zagier (Mathematics Professor, Max Planck Institute of Mathematics in Bonn e International Centre for Theoretical Physics in Trieste)

Allegro vivace: The Riemann Hypothesis from the point of view of Statistical Physics

• Giuseppe Mussardo

Adagio ma non troppo: Conjectures and experiments in Number Theory

• Don Zagier

Moderators:

• Marco Andreatta, Geometry Professor – Mathematics Department
• Sandro Stringari, Emeritus Professor – Physics Department.

Attendance:

The event is open to the entire university community. No registration required.

It is possible to attend the meeting in presence for external people after registration at the following email address: seminario.riemann [at] unitn.it subject to availability. “Green Pass Rafforzato” is mandatory.

It is possible also to attend the meeting online through Zoom, after registration  

The Colloquium will be in English. Expected duration: two and a half hours.