Louis-Pierre Arguin (City University of New York and University of Oxford)

Large Deviations of the Riemann Zeta Function and Random Walks

Abstract

In this talk, I will present a proof that the measures of level sets of the Riemann zeta function have Gaussian tail, up to a constant C, for values in suitable regimes. The lower bound is unconditional whereas the upper bound necessitates the Riemann hypothesis for high enough values. As a corollary, we recover the best-known bounds on the moments on the critical line. The proof relies on the recursive scheme of prior work with Bourgade & Radziwill that is inspired by a random walk heuristic. The upper bound also combines ideas of Soundararajan and Harper. We will discuss possible improvements to the constant C as well as the connections with the Keating-Snaith Conjecture from Random Matrix Theory for the optimal constant.

This is joint work with Emma Bailey & Asher Roberts and Nathan Creighton.

Emma Bailey (University of Bristol)

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Paul Bourgade (Courant Institute, NYU)

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Régis de la Breteche (Université Paris Cité)

The Central Limit Theorem for extended Rademacher random multiplicative random with polynomial phase
Abstract
We will  present some news results on   extended Rademacher random multiplicative random with polynomial phase. This a joint work still in progress with Victor Wang and  Max Xu.

Hung Bui (University of Manchester)

Weighted central limit theorem for central values of L-functions.

Abstract

A classical result of Selberg says that log|zeta(1/2 + it)| has a Gaussian limit distribution. We expect the same thing holds for log|L(1/2, chi)| for chi ranging over the primitive Dirichlet characters modulo q, as q tends to infinity. Proving such a result remains completely out of reach, as it would imply that 100% of these central L-values are non-zero, which is a well-known open conjecture. In this talk, I will describe how one can establish a weighted central limit theorem for the central values of Dirichlet L-functions. Under the Generalized Riemann Hypothesis, one can also obtain a weighted central limit theorem for the joint distribution of the central L-values corresponding to twists of two distinct primitive Hecke eigenforms. This is joint work with Natalie Evans, Stephen Lester and Kyle Pratt.

Vorrapan Chandee (Kansas State University)

The sixth moment of an orthogonal family of L-functions
Abstract

I will discuss an asymptotic formula for the sixth moment of a large orthogonal family of L-functions, associated with holomorphic Hecke newforms of level $q$, averaged over $q \asymp Q$. This is based on my on-going joint work with S.Baluyot and X.Li.

Cécile Dartyge (Université de Lorraine)

Powerfree ellipsephic integers

Abstract

Let $b\in\mathbb$, $b\ge 3$ and $\mathcal\subsetneq\$. The ellipsephic integers or integers with missing digits, are the integers with all their base-$g$ digits in $\mathcal$.In this talk we will present some results on the existence of powerfree ellipsephic integers in the case of subsets $\mathcal$ of small cardinality.

\noindent This is joint work with Anne de Roton and Thomas Stoll.

Sary Drappeau (Université Clermont-Auvergne and Institut Universitaire de France)

Self-intersections points of expanding horocycles

Abstract
The expanding horocycle H(y) = \ \subset \operatorname(2, \mathbb) \backslash \mathbb$ intersects itself at about y^ points, as y\to 0. This talk will concern various aspects of the distribution of these points and the arcs they delimit, with a few results and many questions. This is joint work with Min Lee.

Alexandra Florea (UC Irvine)

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Kevin Ford (U Illinois, Urbana-Champaign)

Prime gaps, Hardy-Littlewood conjectures, the interval sieve and random models for primes 

Abstract

We'll discuss connections between the distribution of primes in short intervals, gaps between consecutive primes, the Hardy-Littlewood k-point correlation conjectures, extremal behavior of interval sieves and random models for primes (Cramer, Granville, Banks-Ford-Tao).

Leo Goldmakher (Williams College)

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Ofir Gorodetsky (Technion – Israel Institute of Technology)

The distribution of partial sums of the Steinhaus function

Abstract
The Steinhaus function is a random, completely multiplicative function on the integers, whose values on primes are iid random variables uniformly distributed on the complex unit circle. Its study is motivated by the study of 'oscillatory' multiplicative functions such as Dirichlet characters.
We'll describe recent joint work with Mo Dick Wong, where the limiting distribution of the partial sums of the Steinhaus function was determined. The limiting distribution is Gaussian with random variance; the variance is given by the total mass of a random measure. This measure is an instance of critical multiplicative chaos. We'll explain the result and highlight the key ideas of the proof.

Oleksiy Klurman (University of Bristol)

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Emmanuel Kowalski (ETH Zürich)

Spectrally indistinguishable pseudorandom graphs (joint works with A. Forey, J. Fresán and Y. Wigderson)

Abstract

Many properties of finite graphs have a well-defined statistical behavior in various models of random graphs. It may be very difficult however to find explicit examples of graphs which display these properties. We show that equidistribution theorems for various exponential sums lead to many deterministic examples of graphs with eigenvalues distributed according to the semicircle distribution, and that such examples can even be extremal for certain Ramsey-theoretic properties.

Valeriya Kovaleva (CRM Montréal)

On integers divisible by a shifted prime in a given interval

Abstract
Let H(x,y,z) be the number of integers with a divisor in a given interval (y,z]. Counting such integers is closely related to the multiplication table problem. In 2008, Ford found the order of magnitude of  H(x,y,z) in all ranges of interest. In this talk we consider a variant of this problem, where the divisors are restricted to shifted primes. We observe multiple phase transitions in the behaviour of the adjusted function as well as the anatomy of integers possessing shifted prime divisors in a prescribed interval depending on its logarithmic length. This is joint work with R. Abi Abdallah, J. Schlitt, and N. Tardy.

Youness Lamzouri (Université de Lorraine and Institut Universitaire de France)

Real zeros of L'(s, \chi_d) and the Baker-Montgomery conjecture

Abstract

In 1990, R. C. Baker and H. L. Montgomery conjectured that for almost all fundamental discriminants $d$, the derivative of the Dirichlet L-function associated to the quadratic character modulo d has around \log\log |d| real zeros on the interval [1/2, 1]. Baker and Montgomery's motivation in studying these zeros stems from their connection to real zeros of Fekete polynomials and to sign changes of real character sums. In this talk I will present recent work that settles this conjecture (up to a small logarithmic factor of \log\log\log |d|). This is based on a joint work with Oleksiy Klurman and Marc Munsch for the lower bound, and a more recent work joint with Kunjakanan Nath for the upper bound.

Junxian Li (UC Davis)

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James Maynard (University of Oxford)

Exponential sums over primes

Abstract
A classical result of Vinogradov shows that the exponential sum over primes \(\sum_e(\alpha p)\) is bounded by \(x/q^ + x^\) (up to an \(x^\) factor), whenever \(\alpha \) has the Diophantine approximation \(a/q+O(1/q^2)\) for some \(q
This has resisted improvements for the past 80 years (beyond refinements to the \(x^\)).
The \(x/q^\) term cannot be improved without a breakthrough on our understanding of Siegel zeros.
I'll discuss how new methods allow us to improve the \(x^\) term to \(x^\), which in turn should have various applications to additive problems related to the primes.

Sarah Peluse (Stanford University)

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Maksym Radziwill (Northwestern University)

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Joël Rivat (Université d'Aix-Marseille et Institut Universitaire de France)

Prime numbers with an almost prime reverse

Abstract

Given an integer $b \geq 2$, we define the \emph in base $b$ of an integer $n \geq 0$ to be the integer obtained by reversing the digits of $n$. The existence of infinitely many prime numbers whose reverse is also prime is an open problem which seems at least as difficult as the twin prime conjecture.  In a joint work with Cécile Dartyge and Cathy Swaenepoel, we show that there are infinitely many prime numbers with an almost prime reverse. More precisely, we show that there exist an explicit $\Omega_b \in \mathbf$ and $c_b > 0$ such that, for at least $c_b b^\lambda \lambda^$ primes $p \in \left[ b^, b^\lambda \right]$, the reverse of $p$ has at most $\Omega_b$ prime factors. The proof is based on sieve methods and a result we obtain in the spirit of the Bombieri-Vinogradov theorem concerning the distribution in arithmetic progressions of the reverse of prime numbers.

Brad Rodgers (Queen's University)

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Fernando Xuancheng Shao (University of Kentucky)

Linear equations in Piatetski-Shapiro primes

Abstract

We establish discorrelation estimates between the Piatetski-Shapiro primes (primes of the form ⌊n^c⌋) and arbitrary nilsequences, where c > 1 is sufficiently close to 1. This extends earlier works which treated linear or polynomial exponential phase functions and provides the first higher-order uniformity result for Piatetski-Shapiro primes. As a consequence, we obtain an asymptotic formula for linear equations in Piatetski-Shapiro primes, thereby generalizing the Green-Tao theorem on linear equations in primes to this sparse setting. This is joint work with Yu-Chen Sun.

K. Soundararajan (Stanford University)

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Joni Teräväinen (University of Cambridge)

Linnik's problem for the Möbius function

Abstract
In 1944, Linnik showed that the least prime in an arithmetic progression is bounded polynomially in terms of the modulus of the progression. Since then, many works have focused on improving the exponent in the polynomial dependence. In this talk we consider an analogue of Linnik's problem for the Möbius function and prove that for this variant the exponent can be taken to be 2. This is based on joint work with Kaisa Matomäki.

Mo Dick Wong (Hong Kong University)

Universality of critical multiplicative chaos

Abstract
The problem of universality in multiplicative chaos asks whether the limiting random measure depends only on the underlying log-correlated field, rather than on the particular regularisation through which the measure is constructed. This question arises naturally in recent joint work with Ofir Gorodetsky, where we need to show that different approximations of random Euler products lead to the same chaos measure in order to apply a martingale CLT to the study of partial sums of a Steinhaus multiplicative function. A key technical challenge at criticality is the lack of sufficient integrability to apply a direct second moment argument. In this talk, I will explain a modified approach that avoids the use of barrier events, relying instead on change-of-measure and coupling arguments.

Asif Zaman (University of Toronto)

Effective Brauer–Siegel theorems for Artin L-functions

Abstract

Given a number field K, in a now classic work, Stark pinpointed the possible source of a so-called Landau–Siegel zero of the Dedekind zeta function of K and used this to give effective upper and lower bounds on its residue at s = 1.

I will discuss an extension of Stark's work giving effective upper and lower bounds for the leading term of the Laurent expansion of general Artin L-functions at s = 1. Up to the value of implied constants, these bounds are as strong as one could reasonably expect given current progress toward the generalized Riemann hypothesis. The bounds are completely unconditional and rely on no unproven hypotheses about Artin L-functions. Strengthened conditional bounds also lead to further questions on the distribution of these values.

This is joint work with Peter Cho and Robert Lemke Oliver.