Tuesday March 25th---Thursday March 27th 2025

A three day workshop at the University of Warwick, bringing together number theorists and probabilists.

About this workshop

Multiplicative chaos is a fundamental object in probability and mathematical physics, dating back to work of Kahane in the 1980s. In the last decade or so, our understanding of multiplicative chaos has greatly expanded, especially in the delicate case of so-called critical chaos. It has also been realised that there are very close connections between this object and fundamental problems in analytic number theory, including the value distribution of the Riemann zeta function and L-functions; the size of Dirichlet character sums; and many questions involving random multiplicative functions.

The goal of this workshop is to bring together number theorists and probabilists interested in this exciting interface, including early career mathematicians and others fairly new to the area, to share recent progress and open questions.

Invited speakers

Louis-Pierre Arguin (Oxford)

Emma Bailey (Bristol) TBC

Ofir Gorodetsky (Technion, Haifa)

Oleksiy Klurman (Bristol)

Youness Lamzouri (Lorraine)

Joseph Najnudel (Bristol)

Ellen Powell (Durham)

Eero Saksman (Helsinki)

Victor Wang (IST Austria)

Mo Dick Wong (Durham)

Max Xu (Courant Institute, New York)

Talk titles and abstracts

• Louis-Pierre Arguin: Moments of the Riemann zeta function in short intervals
• I will discuss recent progress in estimating the moments of the zeta function in short intervals, i.e., intervals whose length grows or shrinks slowly going up the critical line. I will discuss the techniques that were first developed to estimate the maximum on such intervals in relation to the Fyodorov-Hiary-Keating Conjecture. These tools originated from the study of branching random walks and log-correlated processes. The connection with Gaussian Multiplicative Chaos will also be discussed. This is based on joint works with E. Bailey, C. Chang, and J. Hamdan, and also P. Bourgade and M. Radziwill.
• Ofir Gorodetsky: On twisted sums of random multiplicative functions
• Let alpha be a Steinhaus multiplicative function. The distribution of the normalised sum of alpha(n) over integers up to x is not known, although there has been considerable work on its moments by Harper which shows connection to critical multiplicative chaos. In this talk, which is based on joint work with Mo Dick Wong, we will explain how one can obtain the limiting distribution of a somewhat sparser sum, namely the normalised sum of alpha(n) f(n) where f is a function such that the mean value of |f(p)|^2 is less than 1. The proof applies the martingale CLT and then relates the associated bracket process to a subcritical multiplicative chaos measure.
• Oleksiy Klurman: Exponential sums and multiplicativity
• I will talk about recent results aiming to understand the exponential sums \sum_{n\le x}f(P(n))e(n\theta) for integer valued polynomial P. Specified to f being random multiplicative function or a Dirichlet character, this led to some interesting consequences.
• Youness Lamzouri: Sign changes of character sums and random multiplicative functions
• In this talk, I will present a simple and efficient method, which has its roots in the work of Baker and Montgomery, for producing sign changes of weighted sums of certain real multiplicative functions. I will then illustrate two applications to sums of quadratic Dirichlet characters and sums of random Rademacher multiplicative functions. This is a joint work with O. Klurman and M. Munsch.
• Joseph Najnudel: The Fourier coefficients of the holomorphic multiplicative chaos
• In this talk, we consider the coefficients of the Fourier series obtained by exponentiating a logarithmically correlated holomorphic function on the open unit disc, whose Taylor coefficients are independent complex Gaussian variables, the variance of the coefficient of degree k being theta/k where theta > 0 is an inverse temperature parameter. In joint articles with Paquette, Simm and Vu, we show a randomized version of the central limit theorem in the subcritical phase theta < 1, the random variance being related to the Gaussian multiplicative chaos on the unit circle. We also deduce, from results on the holomorphic multiplicative chaos, other results on the coefficients of the characteristic polynomial of the Circular Beta Ensemble, where the parameter beta is equal to 2/theta. In particular, we show that the central coefficient of the characteristic polynomial of the Circular Unitary Ensembles tends to zero in probability, answering a question asked in an article by Diaconis and Gamburd.
• Ellen Powell: An overview of Critical Gaussian multiplicative chaos
• This talk will focus on the construction of Gaussian multiplicative chaos measures when the associated parameter is “critical”; that is, where the usual approximation procedure breaks down. I will aim to give a gentle and self-contained introduction to the theory, and discuss some recent results and open questions.
• Eero Saksman: On the functional statistics of the Riemann zeta function on the critical line
• We will discuss some (old and hopefully also new) results on the statistics of the Riemann zeta function on short intervals on the critical line. At the beginning we will recall Gaussian multiplicative chaos (GMC), especially the critical GMC. The talk is based on collaboration with Adam Harper (Warwick) and Christian Webb (Helsinki).
• Victor Wang: Average sizes of mixed character sums
• In joint work with Max Xu, we prove that the average size of a certain smoothly weighted, mixed character sum of length x is on the order of sqrt(x), under a weak Diophantine genericity condition on the angle \theta of the additive character e(n\theta). Certain quadratic Diophantine equations play a key role. In contrast, it was proved by Harper that the average size is o(sqrt(x)) for rational \theta. Some remaining open questions will be highlighted.
• Mo Dick Wong: On subcritical non-Gaussian multiplicative chaos
• We consider two approximation schemes for the construction of non-Gaussian multiplicative chaos associated to a class of random Dirichlet series, and show in the entire subcritical regime that they give rise to the same limiting measure that describes the asymptotic distribution of partial sums of random multiplicative functions. Our approach uses a modified second moment method with the help of a new coupling argument, and does not rely on any Gaussian approximation or thick point analysis. This is a joint work with Ofir Gorodetsky.
• Max Xu: Distribution of random multiplicative functions over short intervals
• I will discuss limiting distribution of partial sums of random multiplicative functions in short intervals $[x, x+y]$. With a suitable scaling factor, we show that the limiting distribution is Gaussian as long as $y=o(x)$. The interesting feature is that there is a transition point where the scaling factor differs from the usual normalization factor $1/\sqrt{y}$. This is based on joint work in progress with Adam Harper and Kannan Soundararajan.
• Seth Hardy: The distribution of random multiplicative functions with a large prime factor
• It was first observed by Harper in 2020 that the partial sums of random multiplicative functions are related to critical Gaussian multiplicative chaos, and consequently that their typical size is significantly smaller than their standard deviation. This is also true when one restricts the sum to integers that have a large prime factor. I will discuss recent progress towards understanding the distribution of these restricted sums.
• Ethan Lee: Unconditional progress toward an infamous conjecture of Legendre
• Legendre conjectured that there is always a prime between consecutive perfect squares $n^2$ and $(n+1)^2$. Sorenson and Webster recently verified that Legendre's conjecture is true for all integers $n < 2^{46}$, but theoretical methods cannot prove the result (even under the assumption of the Riemann Hypothesis) for larger $n$. So, it is natural to consider a related problem instead: what is the least integer $k$ such that there is always a prime between consecutive perfect $k$th powers $n^k$ and $(n+1)^k$? Assuming the Riemann Hypothesis, we can prove the result for $k = 3$ (i.e., cubes), and without the Riemann Hypothesis, we can only prove the result for $k = 90$ (this is recent work by Cully-Hugill and Johnston). In this talk, I will describe the stumbling blocks that prevent us from doing better than $k = 90$ without assuming the Riemann Hypothesis, and introduce a small modification to the problem that will enable us to prove an almost-complete result for cubes without assuming the Riemann Hypothesis.
• Andrew Pearce-Crump: Number Theory versus Random Matrix Theory: the joint moments story
• It has been known since the 80s, thanks to Conrey and Ghosh, that the average of the square of the Riemann zeta function, summed over the extreme points of zeta up to a height T, is $\frac{1}{2} (e^2-5) \log T$ as $T \rightarrow \infty$. This problem and its generalisations are closely linked to evaluating asymptotics of joint moments of the zeta function and its derivatives, and for a time was one of the few cases in which Number Theory could do what Random Matrix Theory could not. RMT then managed to retake the lead in calculating these sorts of problems, but we may now tell the story of how Number Theory is fighting back, and in doing so, describe how to find a full asymptotic expansion for this problem, the first of its kind for any nontrivial joint moment of the Riemann zeta function. This is joint work with Chris Hughes and Solomon Lugmayer.
• Arshay Sheth: Euler products at the central point
• Even though Euler products of L-functions are generally only valid to the right of the critical strip, there is a strong sense in which they should persist even inside the critical strip. For instance, a random Euler product obtained from Steinhaus or Rademacher random multiplicative functions converges almost surely when Re(s)>1/2 and in fact, for entire L-functions, the convergence of the Euler product inside the right half of the critical strip is known to be equivalent to the Generalized Riemann Hypothesis for the L-function. The behaviour of Euler products on the critical line (in particular, at the central point) is more delicate and its expected behaviour is not consistent with the prediction from the random models. In this talk, I will give a brief introduction to previous results on this topic and then discuss my recent work on establishing an asymptotic formula, conditional on GRH and outside a set of finite logarithmic measure, for partial Euler products of entire L-functions at the central point of the critical strip.

Schedule

All talks will take place in room B3.02, on the top floor of the Zeeman Building at the University of Warwick.

Tuesday March 25th

10.30am-11.00am: Welcome tea/coffee

11.00am-12.00pm: Louis-Pierre Arguin

12.00pm-12.30pm: Arshay Sheth

12.30pm-2.00pm: Lunch

2.00pm-3.00pm: Ellen Powell

3.00pm-3.30pm: Tea/coffee

3.30pm-4.30pm: Ofir Gorodetsky

Wednesday March 26th

9.30am-10.30am: Mo Dick Wong

10.30am-11.00am: Tea/coffee

11.00am-12.00pm: Victor Wang

12.00pm-12.30pm: Seth Hardy

12.30pm-2.00pm: Lunch

2.00pm-3.00pm: Max Xu

3.00pm-3.30pm: Tea/coffee

3.30pm-4.30pm: Youness Lamzouri

4.30pm-5.00pm: Andrew Pearce-Crump

Evening: Conference dinner

Thursday March 27th

9.30am-10.30am: Joseph Najnudel

10.30am-11.00am: Tea/coffee

11.00am-12.00pm: Eero Saksman

12.00pm-12.30pm: Ethan Lee

12.30pm-2.00pm: Lunch

2.00pm-3.00pm: Oleksiy Klurman

3.00pm-3.30pm: Goodbye tea/coffee

Registration

Registration for this workshop has closed.

Practicalities

Tea/coffee and lunch will be provided each day for registered participants, and a conference dinner will be held on the evening of Wednesday 26th.

You can find more details about getting to the University of Warwick campus, and the Zeeman Building, at https://warwick.ac.uk/fac/sci/maths/research/mrc/visit/Link opens in a new window

Unfortunately, we cannot arrange accommodation for workshop participants (except for invited speakers, who will have received separate details). Some local accommodation options include:

The Village Hotel, Coventry:

https://www.village-hotels.co.uk/coventryLink opens in a new window

Holiday Inn, Kenilworth:

https://www.guestreservations.com/holiday-inn-kenilworth-warwick/booking?s=aboutthehotel&gclid=EAIaIQobChMIoe3G2oHA_QIV54BQBh07pAzuEAAYASACEgJrXPD_BwELink opens in a new window

The Abbey Field, Kenilworth:

Abbey Field Pub Restaurant in Kenilworth, (chefandbrewer.com)Link opens in a new window

The Windmill Village, Coventry:

http://www.windmillvillagehotel.co.uk/contact-us/Link opens in a new window

Contact

If you have any queries about this workshop, please contact Prof. Adam Harper: a.harper@warwick.ac.uk

Funding acknowledgement

We gratefully acknowledge support from EPSRC grant number EP/V055755/1, and from the MRC at the University of Warwick.