Simons Foundation Launches Collaboration on Universal Statistics in Number Theory

The new collaboration will explore randomness and correlations in number theory, tackling problems related to the Riemann zeta function, prime numbers and other major questions in the field.

An abstract 3D Multi Сolored Flowing Seamless Pattern representative of a Gaussian multiplicative chaos plot. — oxygen

Mathematicians have long recognized that randomness appears in number theory. For instance, while we know there are infinitely many prime numbers, mathematicians cannot predict exactly when the next prime will pop up as numbers increase. This apparent randomness includes some subtle correlations. For example, the famous twin prime conjecture — formalized in the 1840s — posits that there are an infinite number of prime pairs two numbers apart (such as 11 and 13). Today, breakthroughs in analytic number theory are finally allowing mathematicians to make strides in quantifying the hidden patterns in the randomness.

The new Simons Collaboration on Universal Statistics in Number Theory will apply modern developments in statistical physics to uncover new connections across the field, one of the oldest disciplines in mathematics.

“The heart of our collaboration is to make progress on a collection of problems in number theory using sophisticated models of correlated randomness,” says Collaboration Director Jon Keating, a professor at the University of Oxford. “It’s a surprising idea that you can often use probability to understand problems in number theory, since it seems to be about exact questions, but many problems in number theory have a flavor of randomness to them.”

The Riemann zeta function, which dates to the mid-19th century, encodes information about the primes and reflects that balance between randomness and correlation. The collaboration will apply new methods to examine the bounds of the Riemann zeta function, which oscillates chaotically in the area of interest for the famous Riemann hypothesis. Solving that hypothesis is expected to give deep new insights into the prime numbers.A recent breakthrough came from statistical physics and a concept called Gaussian multiplicative chaos (GMC), which measures fractal chaos. Number theorists realized that the Riemann zeta function reflects properties of GMC. A flurry of conjectures and proofs arose from that realization. The collaboration will translate that momentum into an enduring area of research. Long-term conjectures relate the Riemann zeta function to random matrices and other well-studied constructions, and, as Keating says, “a whole new world has opened up that we weren’t expecting.”

Simons Collaborations in Mathematics and the Physical Sciences bring together groups of outstanding researchers to address topics of fundamental scientific importance. Collaborations receive up to $2 million per year for an initial period of four years, including indirect costs, and may be extended for an additional three years. The collaboration will be funded by grants from Simons Foundation International administered by the Simons Foundation.

The members of the new collaboration are:

Jon Keating
Director; University of Oxford

Louis-Pierre Arguin
PI; Baruch College

Paul Bourgade
PI; New York University

Adam Harper
PI; University of Warwick

Dimitris Koukoulopoulos
PI; University of Montreal

Kaisa Matomäki
PI; University of Turku

James Maynard
PI; University of Oxford

Maksym Radziwill
PI; New York University

Peter Sarnak
PI; Institute for Advanced Study and Princeton University

Will Sawin
PI; Princeton University

Kannan Soundararajan
PI; Stanford University

Melanie Matchett Wood
PI; Harvard University