ViBraNT
Virtual Brazilian Number Theory Seminar
For the continuation of this seminar series go to https://w3.impa.br/~goncalves/vibrant.html
Past edition
Organizers: Marco Aymone (UFMG), André Contiero (UFMG), Ramon Nunes (UFC) and Gugu (IMPA)
YouTube Channel: Most of our talks are in YouTube in the following link
Past talks
2021
June 17 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Ricardo Misturini (UFRGS)
Title: Law of the Iterated Logarithm for a Random Dirichlet Series
Abstract: We consider the random Dirichlet series F(σ) obtained when, in each term of the sum that defines the Riemann Zeta function ζ(σ), we put + or - signs chosen independently and uniformly at random. This series converges when σ > 1/2. We study the behavior of F(σ) when σ goes to 1/2, providing a Law of the Iterated Logarithm, which describes the magnitude of the fluctuations of F(σ). This is a joint work with Marco Aymone and Susana Frómeta.
June 10 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Anders Södergren (Chalmers)
Title: Can a random lattice and its dual be independent?
Abstract: In this talk I will discuss Rogers' mean value formula in the space of unimodular lattices as well as a recent generalization of Rogers' formula. In particular, I will describe a formula for mean values of products of Siegel transforms with arguments taken from both a lattice and its dual lattice. The main application is a result on the joint distribution of the vector lengths in a random lattice and its dual lattice in the limit as the dimension of the lattices tends to infinity, and provides a partial affirmative answer to the question in the title. This is joint work with Andreas Strömbergsson.
June 03 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Vorrapan Chandee (KSU)
Title: The sixth moment of Dirichlet L-functions without average in the t-aspect
Abstract: We prove an asymptotic for the sixth moment of Dirichlet L-functions averaged over primitive characters modulo q, over all moduli q <= Q. Unlike the previous work of Conrey, Iwaniec, and Soundararajan, we do not need to include an average on the critical line, thus requiring treatment of the "unbalanced" sums. This is a joint work with Xiannan Li, Kaisa Matomaki, and Maksym Radziwill.
May 27 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Louis Chen (National University of Singapore)
Title: A probabilistic approach to the Erdös-Kac theorem for additive functions
Abstract: We present a new approach to assessing the rates of convergence to the Gaussian and Poisson distributions in the Erdös-Kac theorem for additive arithmetic functions of a random integer. Our approach is probabilistic, working directly on spaces of random variables without any use of Fourier analytic methods. Of the methods we used is Stein’s method. Our results generalize the existing ones in the literature. This talk is based on joint work with Arturo Jaramillo and Xiaochuan Yang.
May 20 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Sávio Ribas (UFOP)
Title: Some direct and inverse zero-sum problems
Abstract: In this talk, we will introduce the main zero-sum problems in additive combinatorics. In particular, we will define the Davenport and the Erdös-Ginzburg-Ziv constants, among other similar constants for finite groups. We will also present their main results so far and Gao's conjecture that connects some of these constants (which has already been proven for abelian groups). In addition, we will present the similar weighted problems and the inverse problems. This is a joint work with D.V. Avelar, F.E. Brochero Martínez, A. Lemos and B.K. Moryia.
May 13 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Daniel Fiorilli (CNRS)
Title: Higher moments of primes in intervals and in arithmetic progressions
Abstract: Since the work of Selberg and of Barban, Davenport and Halberstam, the variances of primes in intervals and in arithmetic progressions has been widely studied and continue to be an active topic of research. However, much less is known about higher moments. Hooley established a bound on the third moment in progressions, which was later sharpened by Vaughan for a variant involving a major arcs approximation. Little is known for moments of order four or higher, other than the conjecture of Hooley and the conditional result of Montgomery-Soundararajan. In this talk I will discuss recent joint work with Régis de la Bretèche on weighted moments in short intervals and on weighted moments of moments in progressions. In particular we will show how to deduce sharp unconditional omega results on all weighted even moments in certain ranges.
May 06 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Youness Lamzouri (Institut Elie Cartan de Lorraine) [video]
Title: Zeros of linear combinations of L-functions near the critical line
Abstract: In this talk, I will present a recent joint work with Yoonbok Lee, where we investigate the number of zeros of linear combinations of $L$-functions in the vicinity of the critical line. More precisely, we let $L_1, \dots, L_J$ be distinct primitive $L$-functions belonging to a large class (which conjecturally contains all $L$-functions arising from automorphic representations on $\text{GL}(n)$), and $b_1, \dots, b_J$ be real numbers. Our main result is an asymptotic formula for the number of zeros of $F(\sigma+it)=\sum_{j\leq J} b_j L_j(\sigma+it)$ in the region $\sigma\geq 1/2+1/G(T)$ and $t\in [T, 2T]$, uniformly in the range $\log \log T \leq G(T)\leq (\log T)^{\nu}$, where $\nu\asymp 1/J$. This establishes a general form of a conjecture of Hejhal in this range. The strategy of the proof relies on comparing the distribution of $F(\sigma+it)$ to that of an associated probabilistic random model.
April 22 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Chantal David (Concordia University)
Title: One-Level density for cubic characters over the Eisenstein field
Abstract: We show that the one-level density for $L$-functions associated with the cubic residue symbols $\chi_n$, with $n \in \mathbb{Z}[\omega]$ square-free, satisfies the Katz-Sarnak conjecture for all test functions whose Fourier transforms are supported in $(-13/11, 13/11)$, under GRH. This is the first result extending the support outside the trivial range $(-1, 1)$ for a family of cubic $L$-functions. This implies that a positive proportion of the $L$-functions associated with these characters do not vanish at the central point $s = 1/2$. A key ingredient is a bound on an average of generalized cubic Gauss sums at prime arguments, whose proof is based on the work of Heath-Brown and Patterson.
April 15 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Brad Rodgers (Queen’s University) [video]
Title: The distribution of random polynomials with multiplicative coefficients
Abstract: A classic paper of Salem and Zygmund investigates the distribution of trigonometric polynomials whose coefficients are chosen randomly (say +1 or -1 with equal probability) and independently. Salem and Zygmund characterized the typical distribution of such polynomials (gaussian) and the typical magnitude of their sup-norms (a degree N polynomial typically has sup-norm of size $\sqrt{N \log N}$ for large N). In this talk we will explore what happens when a weak dependence is introduced between coefficients of the polynomials; namely we consider polynomials with coefficients given by random multiplicative functions. We consider analogues of Salem and Zygmund's results, exploring similarities and some differences.
Special attention will be given to a beautiful point-counting argument introduced by Vaughan and Wooley which ends up being useful.
This is joint work with Jacques Benatar and Alon Nishry.
April 08 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Marco Aymone (UFMG) [video]
Title: Some oscillation theorems in analytic and probabilistic Number Theory
Abstract: This talk will be divided into two independent parts. In the first part of the talk I will discuss the prime number race mod 4: Usually one assumes standards conjectures as GRH to deduce some results that captures the intuition behind the Tchébyshev bias -- I will do the other way around. In the second part of the talk I will discuss a recent work with Winston Heap and Jing Zhao on sign changes of the partial sums of a random multiplicative function.
March 25 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Chris Hughes (University of York) [video]
Title: A Random Matrix Model for Gram's Law
Abstract: It is well known that the counting function for the Riemannzeta zeros, N(T), has a smooth main term and a much smaller discontinuous correction term, S(T). Gram's Law is the observation that between consecutive points where the smooth part of the counting function is an integer, there typically is exactly one zeta zero. This "Law" doesn't hold all the time, and we will use random matrix theory to model the proportion of time the law holds for. The flavour of random matrix theory that normally models the Riemann zeros is the unitary group. However, studying Gram's Law requires the special unitary group, where many of the useful techniques for random unitary matrices fail to hold. Much of this work was done jointly with my former PhD student Catalin Hanga.
March 18 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Daniele Mastrostefano (Warwick) [video]
Title: The partial sum of a random multiplicative function on integers with a large prime factor
Abstract: Let $f(n)$ be a Rademacher random multiplicative function. We prove that, for any $\eps>0$ and as $x\rightarrow +\infty$, we almost surely have
$$\sum_{\substack{n\leq x\\ P(n)>\sqrt{x}}}f(n)\ll\sqrt{x}(\log\log x)^{1/4+\eps},$$
where $P(n)$ stands for the largest prime factor of $n$.
This is close to be sharp and gives an indication of the size of the largest fluctuations of the full partial sum.
March 11 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Sandro Bettin (University of Genova)
Title: Modularity and distribution of quantum knots invariants
Abstract: We consider Zagier's modularity conjecture for the colored Jones polynomials of hyperbolic knots. We prove this conjecture in some cases and show that, in the case of the 4_1 knot, one can also deduce a law of large for the values of the colored Jones polynomial at roots of unity. This is joint work with Sary Drappeau.
February 25 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Caroline Turnage-Butterbaugh (Carleton College) [video]
Title: Gaps between zeros of the Riemann zeta-function
Abstract: Let $0 < \gamma_1 \le \gamma_2 \le \cdots $ denote the ordinates of the complex zeros of the Riemann zeta-function function in the upper half-plane. The average distance between $\gamma_n$ and $\gamma_{n+1)$ is $2\pi / \log \gamma_n$ as $n\to \infty$. An important goal is to prove unconditionally that these distances between consecutive zeros can much, much smaller than the average for a positive proportion of zeros. We will discuss the motivation behind this endeavor, progress made assuming the Riemann Hypothesis, and recent work with A. Simonič and T. Trudgian to obtain an unconditional result.
February 18 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Andreas Weingartner (South Utah university)
Title: An extension of the Siegel-Walfisz theorem
Abstract: We extend the Siegel-Walfisz theorem to a family of integer sequences that are characterized by constraints on the size of the prime factors. Besides prime powers, this family includes smooth numbers, almost primes and practical numbers.
February 11 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Joni Teräväinen (Oxford)
Title: Higher order uniformity of the Möbius function
Abstract: I will discuss recent work where we prove that the Möbius function is orthogonal to a wide class of phase functions (including all polynomial phases) on almost all very short intervals. I will also discuss applications to superpolynomial word complexity for the Liouville sequence and to a new averaged version of Chowla's conjecture. This is joint work with Kaisa Matomäki, Maksym Radziwiłł,Terence Tao and Tamar Ziegler.
February 04 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Marc Munsch (Graz)
Title: Pair correlation of sequences: metric results and a modified additive energy
Abstract: The uniform distribution of a sequence $\{x_n\}_{n\geq 1}$ measures the pseudo-random behavior at a global scale. At a more localized scale, we can study the pair correlation for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form $(a_n \alpha)_{n \geq 1}$ has been pioneered by Rudnick, Sarnak and Zaharescu. Recently, a general framework was developed which gives a criterion for Poissonian pair correlation of such sequences for almost $\alpha \in (0,1)$, in terms of the additive energy of the integer sequence $\{a_n\}_{n \geq 1}$. In the present talk we will discuss a similar framework in the more delicate case where $\{a_n\}_{n \geq 1}$ is a sequence of reals. We give a criterion involving a modified version of the additive energy expressed via a diophantine inequality. We give several concrete applications of our method and present some open problems. This is joint work with Christoph Aistleitner and Daniel EL-Baz.
January 28 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Kyle Pratt (Oxford) [video]
Title: Landau-Siegel zeros and central values of L-functions
Abstract: Researchers have tried for many years to eliminate the possibility of Landau-Siegel zeros---certain exceptional counterexamples to the Generalized Riemann Hypothesis. Often one thinks of these zeros as being a severe nuisance, but there are many situations in which their existence allows one to prove spectacular, though illusory, results. I will review some of this history and some of these results. In the latter portion of the talk I will discuss recent work, joint with H. M. Bui and Alexandru Zaharescu, in which we show that the existence of Landau-Siegel zeros has implications for the behavior of $L$-functions at the central point.
January 21 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Cathy Swaenepoel (Paris - Diderot)
Title: Prime numbers with preassigned digits
Abstract: Bourgain (2015) estimated the number of prime numbers with a proportion c>0 of preassigned digits in base 2 (c is an absolute constant not specified). We present a generalization of this result in any base $g\geq2$ and we provide explicit admissible values for the proportion c depending on g. Our proof, which adapts, develops and refines Bourgain’s strategy, is based on the circle method and combines techniques from harmonic analysis together with results on zeros of Dirichlet L-functions, notably a very strong zero-free region due to Iwaniec.
January 14 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Olivier Ramaré (Aix-Marseille) [video]
Title: An additive question in multiplicative number theory
Abstract: While studying the representation of a congruence class or a ray-class by a product of three small primes, we stumbled on an auxiliary additive combinatorics question involving sum-free sets in finite abelian groups that seems to be new. The aim of the talk is to present this question.
2020
December 03 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Kevin Ford (University of Illinois at Urbana-Champaign) [video]
Title: Divisors of integers, permutations and polynomials
Abstract: We describe a probabilistic model that describes the statistical behavior of the divisors of integers, divisors of permutations and divisors of polynomials over a finite field. We will discuss how this can be used to obtain new bounds on the concentration of divisors of integers, improving a result of Maier and Tenenbaum. This is joint work with Ben Green and Dimitris Koukoulopoulos.’
November 26 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo) C
Speaker: Sandoel Vieira (IMPA) [video]
Title: M\L is not closed
Abstract: In this talk we will describe joint work with C. G. Moreira, C. Matheus and D. Lima in which we proved that $M\setminus L$ is not a closed subset of $\mathbb{R}$. For that, we show that $1+3/\sqrt{2}$ is a point of the Lagrange spectrum $L$ which is accumulated by a sequence of elements of the complement $M\setminus L$ of the Lagrange spectrum in the Markov spectrum $M$.
November 19 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Kaisa Matomäki (University of Turku) [video]
Title: Almost primes in almost all very short intervals
Abstract: By probabilistic models one expects that, as soon as $h \to \infty$ with $X \to \infty$, short intervals of the type $(x- h \log X, x]$ contain primes for almost all $x \in (X/2, X]$. However, this is far from being established. In the talk I discuss related questions and in particular describe how to prove the above claim when one is satisfied with finding $P_2$-numbers (numbers that have at most two prime factors) instead of primes.
November 12 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Maksym Radziwill (Caltech) [video]
Title: The Fyodorov-Hiary-Keating conjecture
Abstract: I will discuss recent progress on the Fyodorov-Hiary-Keating conjecture on the distribution of the local maximum of the Riemann zeta-function. This is joint work with Louis-Pierre Arguin and Paul Bourgade.
November 05 (Thursday), 14h Brazilian local time (GMT -03:00, America / Sao Paulo)
Speaker: Εfthymios Sofos (University of Glasgow) [video]
Title: Schinzel Hypothesis with probability 1 and rational points
Abstract: Joint work with Alexei Skorobogatov, preprint: https://arxiv.org/abs/2005.02998. Schinzel’s Hypothesis states that every integer polynomial satisfying certain congruence conditions represents infinitely many primes. It is one of the main problems in analytic number theory but is completely open, except for polynomials of degree 1. We describe our recent proof of the Hypothesis for 100% of polynomials (ordered by size of coefficients). We use this to prove that, with positive probability, Brauer–Manin controls the Hasse principle for Châtelet surfaces.
October 29 (Thursday), 14h Brazilian local time (UTC -03:00, America / Sao Paulo)
Speaker: James Maynard (Oxford) [video]
Title: Primes in arithmetic progressions to large moduli
Abstract: I’ll talk about some recent work extending the Bombieri-Vinogradov Theorem to moduli larger than x^{1/2} provided the moduli have a conveniently sized divisor. In different formulations, this allows us to handle moduli as large as x^{3/5}, or allows for complete uniformity with respect to the residue class as in the original Bombieri-Vinogradov theorem.
October 22 (Thursday), 14h Brazilian local time (UTC -03:00, America / Sao Paulo)
Speaker: Thomas Bloom (Cambridge)
Title: Additive structure in dense sets of integers
Abstract: How much additive structure can we guarantee in sets of integers, knowing only their density? The study of which density thresholds are sufficient to guarantee the existence of various kinds of additive structures is an old and fascinating subject with connections to analytic number theory, additive combinatorics, and harmonic analysis. In this talk we will discuss recent progress on perhaps the most well-known of these thresholds: how large do we need a set of integers to be to guarantee the existence of a three-term arithmetic progression? In recent joint work with Olof Sisask we broke through the logarithmic density barrier for this problem, establishing in particular that if a set is dense enough such that the sum of reciprocals diverges, then it must contain a three-term arithmetic progression, establishing the first case of an infamous conjecture of Erdos.
October 15 (Thursday), 14h Brazilian local time (UTC -03:00, America / Sao Paulo)
Speaker: Dimitris Koukoulopoulos (Université de Montréal) [video]
Title: How concentrated can the divisors of a typical integer be?
Abstract: The Delta function measures the concentration of the sequence of divisors of an integer. Specifically, given an integer $n$, we write $\Delta(n)$ for the maximum over $y$ of the number of divisors of $n$ lying in the dyadic interval $[y,2y]$. It was introduced by Hooley in 1979 because of its connections to various problems in Diophantine equations and approximation. In 1981, Maier and Tenenbaum proved that $\Delta(n)>1$ for almost all integers $n$, thus settling a 1948 conjecture due to Erd\H os. In subsequent work, they proved that $(\log\log n)^{c+o(1)}\le \Delta(n)\le (\log\log n)^{\log2+o(1)}$, where $c=(\log2)/\log(\frac{1-1/\log 27}{1-\log3})\approx 0.33827$ for almost all integers $n$. In addition, they conjectured that $\Delta(n)=(\log\log n)^{c+o(1)}$ for almost all $n$. In this talk, I will present joint work with Ben Green and Kevin Ford that disproves the Maier-Tenenbaum conjecture by replacing the constant $c$ in the lower bound by another constant $c’=0.35332277\dots$ that we believe is optimal. We also prove analogous results about permutations and polynomials over finite fields by reducing all three cases to an archetypal probabilistic model.
October 08 (Thursday), 14h Brazilian local time (UTC -03:00, America / Sao Paulo)
Speaker: Sarah Peluse (Oxford) [video]
Title: An asymptotic version of the prime power conjecture for perfect difference sets
Abstract: A subset D of a finite cyclic group Z/mZ is called a “perfect difference set” if every nonzero element of Z/mZ can be written uniquely as the difference of two elements of D. If such a set exists, then a simple counting argument shows that m=n^2+n+1 for some nonnegative integer n. Singer constructed examples of perfect difference sets in Z/(n^2+n+1)Z whenever n is a prime power, and it is an old conjecture that these are the only such n for which a perfect difference set exists. In this talk, I will discuss a proof of an asymptotic version of this conjecture: the number of n less than N for which Z/(n^2+n+1)Z contains a perfect difference set is ~N/log(N).
October 01 (Thursday), 14h Brazilian local time (UTC -03:00, America / Sao Paulo)
Speaker: Lucile Devin (University of Gothenburg)
Title: Chebyshev’s bias and sums of two squares
Abstract: Studying the secondary terms of the Prime Number Theorem in Arithmetic Progressions, Chebyshev claimed that there are more prime numbers congruent to 3 modulo 4 than to 1 modulo 4. We will explain and qualify this claim following the framework of Rubinstein and Sarnak. Then we will see how this framework can be adapted to other questions on the distribution of prime numbers. This will be illustrated by a new Chebyshev-like claim : there are “more” prime numbers that can be written as a sum of two squares with the even square larger than the odd square than the other way around.
September 24 (Thursday), 14h Brazilian local time (UTC -03:00, America / Sao Paulo)
Speaker: Andrés Chirre (NTNU) [video]
Title: The behavior of the argument of the Riemann zeta-function
Abstract: In this talk we will review some recent results related to the argument function of the Riemann zeta function, assuming the Riemann hypothesis. The use of bandlimited approximations and the resonance method will help us to describe the behavior of this oscillatory function. Finally, we will extend these results to the antiderivatives of the argument function that encode, in a certain way, information about the argument function.
September 17 (Thursday), 14h Brazilian local time (UTC -03:00, America / Sao Paulo)
Speaker: Andrew Granville (Université de Montréal) [video]
Title: Heuristics and computations for primes in short intervals; and sieves and Siegel zeros
Abstract: We describe joint work with Allysa Lumley in which we try to get an idea of the range of values the number of primes can take in an interval of length y near to x. Our understanding is limited by our limited understanding of the sieve and, if we have time, we will explain how that understanding cannot be improved without showing that there are no Siegel zeros
September 10 (Thursday), 14h Brazilian local time (UTC -03:00, America / Sao Paulo)
Speaker: Gady Kozma (Weizmann Institute of Science)
Title: Random polynomials, sieves and Dedekind zeta functions
Abstract: What is the probability that a random polynomial with coefficients +/-1 is irreducible over the rationals? This fascinating problem, still open, has seen a lot of progress in the last few years. We will survey this progress, with particular emphasis on new results, joint with Lior Bary-Soroker and Dimitris Koukoulopoulos.
September 03 (Thursday), 14h Brazilian local time (UTC -03:00, America / Sao Paulo)
Speaker: Maxim Gerspach (ETH, Zürich) [video]
Title: Low pseudomoments of the Riemann zeta function and its powers
Abstract: The pseudomoments of the Riemann zeta function are the moments of the partial sums associated to zeta on the critical line. Using probabilistic methods of Harper, we provide bounds which imply the order of magnitude of all pseudomoments. We also provide upper and lower bounds for the pseudomoments of the powers of zeta that are almost-matching when combined with previous bounds of Bondarenko, Heap and Seip, and turn out to behave in a somewhat different manner. In this talk, I will mostly try to give a heuristic argument in support of the results by relating these quantities to moments of random multiplicative functions and to random Euler products.
August 27 (Thursday), 14h Brazilian local time (UTC -03:00, America / Sao Paulo)
Speaker: Oleksiy Klurman (Max Planck and University of Bristol) [video]
Title: Monotone chains in multiplicative sets
Abstract: It is a rather difficult task to show that given a general sequence $a(1),a(2)\dots$ and admissible set of integers $h_1,h_2\dots h_k$ each possible arrangement $a(n+h_1)\le a(n+h_2)\le\dots a(n+h_k)$ occurs for infinitely many integers $n.$. In this talk, we describe how recent advances in multiplicative number theory and theory of automorphic forms allow us to shed some light on such questions related to the coefficients of Hecke cusp forms (based on a joint work with A. Mangerel).
August 20 (Thursday), 14h Brazilian local time (UTC -03:00, America / Sao Paulo)
Speaker: Adam Harper (Warwick)
Title: Multiplicative chaos in number theory
Abstract: Multiplicative chaos is the general name for a family of probabilistic objects, which can be thought of as the random measures obtained by taking the exponential of correlated Gaussian random variables. Multiplicative chaos turns out to be closely connected with various problems in analytic number theory, including the value distribution of the Riemann zeta function on the critical line, the moments of character sums, and various model versions of these problems. I will try to give a gentle introduction to these issues and connections, presenting both results and open problems without assuming too much background knowledge. (This will be a lightly updated version of the talk I gave last year in Cetraro.)
August 13 (Thursday), 14h Brazilian local time (UTC -03:00, America / Sao Paulo)
Speaker: Sam Chow (University of Warwick)
Title: Moments of Weyl sums, restriction estimates, and diophantine equations
Abstract: We discuss the role played by moment estimates for Weyl sums in counting solutions to diophantine equations, and the analogous role played by restriction estimates in the combinatorial theory of diophantine equations. Additionally, we sketch some modern techniques used to prove such estimates.
August 6 (Thursday), 14h: UTC -03:00 (America / Sao Paulo)
Speaker: Winston Heap (Max Planck, Bonn)
Title: Random multiplicative functions and a model for the Riemann zeta function
Abstract: We look at a weighted sum of random multiplicative functions and view this as a model for the Riemann zeta function. We investigate various aspects including its high moments, distribution and maxima.
July 30 (Thursday), 14h: UTC -03:00 (America / Sao Paulo)
Speaker: Jing Zhao (Max Planck, Bonn)
Title: Discrete negative moments of ζ′(ρ)
Abstract: I shall talk about a recent result of a joint work with Winston Heap and Junxian Li. We proved lower bounds for the discrete negative 2kth moments of the derivative of the Riemann zeta function, which agrees with a conjecture of Gonek and Hejhal. We also proved a general formula for the discrete twisted 2nd moment of the Riemann zeta function. This agrees with a conjecture of Conrey and Snaith.
July 23 (Thursday), 14h: UTC -03:00 (America / Sao Paulo) CLOSED
Speaker: Ramon Nunes (UFC, Brazil)
Title: Moments of k-free numbers in arithmetic progressions
Abstract: We will discuss the moments of distribution of $k$-free numbers in arithmetic progressions for which we show estimates improving on previous results by Hall and the author. We will present conjectures due mainly to Montgomery and according to which our results are nearly optimal. The key new idea is to complement Hall’s argument based on the so-called fundamental lemma of Montgomery and Vaughan with some elementary estimates on the region where the previous approach is wasteful.
July 16 (Thursday), 14h: UTC -03:00 (America / Sao Paulo)
Speaker: Alexander Mangerel (CRM, Montreal),
Title: Squarefree Integers in Arithmetic Progressions to Smooth/Friable Moduli
Abstract: I will discuss how to obtain an asymptotic formula (with power-savings error term) for the count of squarefree integers in an arithmetic progression when the modulus does not have any large prime factors, using a blend of cohomological techniques and -adic methods. For this collection of moduli our results go beyond the best existing admissible range obtained recently by Nunes.
This is joint work with C. Perret-Gentil.