About the collaboration
The prime numbers are perfectly determined, yet in many ways they behave as if they were random, and so do the zeros of the Riemann zeta function and rational points on varieties. The simplest random models, which assume complete independence, capture this only to a first approximation and quickly break down. The zeros of the zeta function, for example, do not scatter independently: they repel one another like the eigenvalues of a random matrix.
The collaboration’s guiding thesis is that arithmetic randomness is correlated randomness. The distributions that replace independence are the universal distributions of statistical physics: the eigenvalue statistics of random matrices, Gaussian multiplicative chaos and the Painlevé distributions. Understanding why they appear in number theory is a key step toward progress on central questions such as the twin prime conjecture and the behavior of the Riemann zeta function.
The collaboration brings together experts in analytic number theory, probability and mathematical physics. Its research spans sieve methods and the anatomy of integers, the zeros and large values of L-functions, and the distribution of rational points on varieties: three areas linked by the same probabilistic mechanisms.
This collaboration is part of the Simons Collaborations in Mathematics and Physical Sciences (MPS) program. This work is supported by a Simons Foundation International grant administered by the Simons Foundation.
