MPS Conference on Universal Statistics in Number Theory

The MPS Conference on Universal Statistics in Number Theory brought together researchers working at the interface of analytic number theory, probability, and random matrix theory. Over several days, speakers presented results that illustrate how probabilistic and spectral perspectives are deepening the understanding of primes, L-functions, and algebraic structures. A recurring theme was the growing role of random models and universality principles across different areas of number theory.

All talks reflected very recent developments on these interactions. Vorrapan Chandee verified cases of the Katz–Sarnak conjecture for one-level densities in large orthogonal families of L-functions. Kevin Ford examined probabilistic models for primes and their limitations beyond Cramér-type heuristics. Andrew Granville described optimization problems for multiplicative functions, refining generalizations of Halász’s theorem. Adam Harper introduced ongoing work connecting the square of the Riemann zeta function with critical multiplicative chaos. Jonathan Keating discussed new formulas on joint moments of characteristic polynomials and their relevance for zeta-function derivatives. Dimitris Koukoulopoulos presented a proof of optimal convergence concerning the Poisson–Dirichlet distribution of prime factors of integers. Will Sawin proposed a refined random matrix model for function field L-functions. Victor Wang explored the multiplicative structure within additive Diophantine problems, and Ofer Zeitouni described recent progress on the extremes of characteristic polynomials of circular β-ensembles, making important progress on the Fyodorov–Hiary–Keating conjectures on the random matrix side.

To describe some probabilistic patterns in arithmetic in more detail, we mention a few talks including James Maynard’s, discussing ongoing work on prime detecting sieves, developed jointly with Kevin Ford. His presentation examined how local density heuristics for primes interact with possible global obstructions, and how understanding these obstructions could clarify why many conjectural asymptotics for primes are so difficult to prove.

Maksym Radziwiłł presented progress on joint Linnik problems and Siegel zeros, establishing new forms of equidistribution for embeddings of tori into inner forms of PGL2. The described approach bypasses ergodic methods, and highlights connections between bounds for moments of L-functions and equidistribution problems.

Melanie Matchett Wood spoke on universal distributions of groups and algebraic objects, placing conjectural class group heuristics in a broader setting. She outlined how probabilistic models can describe the distribution of algebraic structures beyond abelian groups, suggesting new directions for understanding universality in arithmetic statistics.

Peter Sarnak concluded the meeting with a survey of distribution of zeros, eigenvalues, and eigenfunctions in number theory. He reviewed how statistical regularities—predicted by random matrix theory and observed in computations—govern both zeta functions and automorphic spectra. The talk emphasized the interplay between numerical patterns, spectral theory, and conjectural models for arithmetic quantities.

Across the talks, several directions emerged: the use of random models to understand arithmetic regularities, new forms of universality across analytic and algebraic settings, and increasing overlap between probabilistic techniques and classical number theory. The meeting reflected a shared interest in unifying analytic, spectral, and probabilistic viewpoints to approach long standing arithmetic questions.

Vorrapan Chandee
Kansas State University

Moments of One-Level Densities for a Large Orthogonal Family of L-functions
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Katz and Sarnak conjectured that the statistics of low-lying zeros of families of L-functions match with the scaling limit of eigenvalues from the random matrix theory. In this talk, I will discuss recent joint work with Yoonbok Lee and Xiannan Li on the nth-centered moments of one level densities of a large orthogonal family of L-functions associated with holomorphic Hecke newforms of level q, averaged over q~Q. The nth centered moments are closely related to the n-level densities of low-lying zeros of L-functions. We verify the Katz–Sarnak conjecture for these statistics, in the range where the sum of the supports of the Fourier transforms of test functions lies in (-4, 4). Key challenges include identifying off-diagonal main terms and resolving a combinatorial problem in matching number-theoretic results with predictions from random matrix theory.
 

Kevin Ford
University of Illinois at Urbana-Champaign

Prime Number Models: Beyond Cramer and Gallagher
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We’ll describe connections between probabilistic models for primes, the Hardy–Littlewood k-tuples conjectures, the distribution of primes in very short intervals, the interval sieve, and hypothetical Landau–Siegel zeros of Dirichlet L-functions. We will emphasize the role and limitations of probabilistic ideas.
 

Andrew Granville
Université de Montréal

Optimization Problems for Multiplicative Functions
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To understand the “usual” behaviour of functions in multiplicative number theory, it is often useful to bound extreme behaviours, Halasz’s theorem being proto-typical. In this talk, we report on several projects on explicit bounds in more general settings, mean values of multiplicative functions whose values at primes are, for example (a) the kth roots of unity or (b) in [-m,m] for fixed m>=1. Also, Halasz’s theorem is known for multiplicative functions that are k-bounded at primes, but the uniformity is poor, and we will see how this can be improved.

This talk includes joint work with Kaisa Matomäki, K. Soundararajan, Kevin Church and Daodao Yang.
 

Adam Harper
University of Warwick

The Square of the Riemann Zeta Function Gives Rise to Critical Multiplicative Chaos
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Multiplicative chaos is a class of random measures that have recently been found to have strong connections with number theoretic objects like L-functions and character sums and the phenomenon of “better than square-root cancellation.” Saksman and Webb have conjectured that integrating test functions against absolute powers of the Riemann zeta function should give rise to these measures. The square of the zeta function is particularly interesting, since this should correspond to the so-called critical chaos. Adam Harper will report on joint work (in preparation) with Saksman and Webb, which proves their conjecture for zeta squared. Harper will try to give a gentle introduction to these problems, and indicate some of the main proof ideas, which may be of independent interest.
 

Jonathan Keating
University of Oxford

Joint Moments
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Jonathan Keating will discuss the evaluation of the joint moments of the characteristic polynomials of random unitary matrices and their derivatives, and in this context the joint moments of the Riemann zeta-function and its derivates, on the critical line. I also hope to discuss extensions to other symmetry classes.
 

Dimitris Koukoulopoulos
University of Montreal

Factoring Random Integers

Billingsley proved that the prime factors of a random integer are known to follow the Poisson–Dirichlet distribution of parameter 1. Arratia gave a new proof of this result by constructing a coupling between the prime factors of a random integer and a Poisson–Dirichlet distribution. His proof furnished a quantitative version of Billingsley’s theorem, which he conjectured could be improved. In this talk, Dimitris Koukoulopoulos will present recent work with Tony Haddad, where we proved Arratia’s conjecture. Furthermore, Koukoulopoulos will explain how to use these methods to deduce various other results about the distribution of divisors of integers.
 

James Maynard
University of Oxford

Prime Detecting Sieves

Many of the most famous open questions on primes are examples of a simple universal statistics phenomenon where we expect the number of primes in a set to be (well-approximated by) a product of local densities of the set. A key reason that proving these guesses appears hard is that there are possible ‘global’ obstructions even for fairly well-understood sets. I’ll talk about joint work with Kevin Ford (in progress) which aims to obtain a better understanding of the nature of these global obstructions.
 

Maksym Radziwill
Northwestern University

Joint Linnik Problems and Siegel Zeros
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We prove the simultaneous equidistribution conjecture of Michel and Venkatesh for the embedding of a torus into two distinct inner forms of PGL_2 assuming a o(1) strengthening of the non-existence of Siegel zeros for quadratic characters.
Since the number of such exceptional characters is essentially O(1) per dyadic interval, our approach improves the previous analytic approach of Blomer–Brumley and the ergodic Aka–Einsiedler–Shapira approach, which covers at best 99% discriminants in [X, 2X] due to a double splitting condition.
The relevance of this topic to the workshop is that an essential tool in the proof are bounds for fractional moments of L-functions; in fact, we uncover a non-trivial intersection with the work of Harper on random multiplicative functions, but we end up not needing it at the end of the day for our specific application (although it would be needed for certain generalizations of our result).
 

Peter Sarnak
Princeton University

Distribution of Zeros, Eigenvalues and Eigenfunctions in Number Theory
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We review some of the statistical laws that appear to govern zeta functions and automorphic forms, and in particular the extent to which they are observed numerically and their impact on extremal properties of arithmetical invariants.
 

Will Sawin
Princeton University

A Refined Random Matrix Model for Function Field L-functions

Will Sawin will explain a proposed model for function field L-functions that combines the random matrix model with Steinhaus random multiplicative functions in a perhaps-unexpected way, and describe how it is possible to calculate some of the statistics of this model, including its moments. The moments agree with predictions made using the Conrey–Farmer–Keating–Rubinstein–Snaith recipe for the moments of the relevant function field L-functions, giving a probabilistic justification for these predictions.
 

Victor Wang
Institute of Mathematics, Academia Sinica

Multiplicative Structure in Additive Problems
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Victor Wang will discuss some Diophantine counting problems for which an analysis via the circle method features the multiplicative statistics of an associated polynomial. Particular focus will be given to a Perazzo cubic fourfold, the subject of joint work in progress with Tim Browning and Ritabrata Munshi.
 

Melanie Matchett Wood
Harvard University

Universal Distributions of Groups and Algebraic Objects

The conjectural distributions for class groups of number fields from the Cohen–Lenstra–Martinet heuristics are known to be universal distributions on abelian groups. We will discuss known results and open problems on the universality of distributions of groups, including non-abelian groups, modules, and other algebraic structures that arise in arithmetic statistics.
 

Ofer Zeitouni
Weizmann Institute of Science

Extrema of the Characteristic Polynomial of C𝛽E Matrices
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The logarithm of the characteristic polynomial of C\(\beta\)E matrices, viewed as a function on the unit circle in the complex plane, is a logarithmically correlated field. Ofer Zeitouni will describe joint work with Elliot Paquette, where we prove that properly centered, the maximum of this field converges to the sum of a Gumbel random variable and an independent random variable \(Z\), thus validating a famous conjecture of Fyodorov–Hiary–Keating. If time permits, Zeitouni will describe recent work identifying the random variable \(Z\).