# SCIENTIFIC PROGRAMS AND ACTIVITIES

September 25, 2017
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th ANNIVERSARY YEAR
 Number Theory Seminar 2012-2013 Fields Institute, Stewart Library(map) Mondays at 3:30 p.m. Organizing Committee: Leo Goldmakher, Youness Lamzouri
 Upcoming Seminars TBA Past Seminars Apr. 8 4:30 p.m. *Please note time change Steve Gonek (University of Rochester) Finite Euler product approximations of the Riemann zeta-function If one could construct a model of the Riemann zeta-function that incorporates its basic properties but has a more transparent structure, this could provide insight into the zeta-function's behavior. I will describe the construction of a family of functions out of finite Euler products and any zeros the zeta-function might have to the right of the critical line. I will then discuss how well these functions approximate the zeta-function both on the Riemann Hypothesis and unconditionally. Apr. 1 Wei Ho (Columbia University) Bounding average ranks for elliptic curves in families In 2010, Bhargava and Shankar found the first unconditional upper bound for the average rank of elliptic curves over Q. We will explain the (quite elementary!) main ideas behind these types of results, namely using orbits of representations to parametrize data related to elliptic curves and then using geometry-of-numbers methods to count the number of relevant orbits. We will also explain generalizations to various families of elliptic curves (joint work with Bhargava). Mar. 25 Carl Pomerance (Dartmouth College) Sums and products What could be simpler than to study sums and products of integers? Well maybe it is not so simple since there is a major unsolved problem: For arbitrarily large numbers N, is there a set of N positive integers where the number of pairwise sums is at most N^{1.99} and likewise, the number of pairwise products is at most N^{1.99}? Erdos and Szemeredi conjecture no. This talk is directed at another problem concerning sums and products, namely how dense can a set of positive integers be if it contains none of its pairwise sums and products? For example, take the numbers that are 2 or 3 mod 5, a set with density 2/5. Can you do better? This talk reports on recent joint work with P. Kurlberg and J. C. Lagarias. Mar. 18 Daniel Fiorilli (University of Michigan) The distribution of the variance of primes in arithmetic progressions Gallagher's refinement of the Barban-Davenport-Halberstam states that V(x;q), the variance of primes up to x in the arithmetic progressions modulo q, is at most x log q, on average over q in the range x/(log x)^A < q < x. It was then discovered by Montgomery that in this range V(x;q) is actually asymptotic to x log q (on average over q); his result was refined by a long list of authors including Hooley, Goldston and Vaughan, and Friedlander and Goldston. Tools used in these papers include the circle method and divisor switching techniques, and under GRH and a strong from of the Hardy-Littlewood Conjecture it is now known that V(x;q) is asymptotic to x log q in the range x^{1/2+o(1)} < q< x. While it is not clear that the asymptotic should hold for more moderate values of q, Keating and Rudnick have proven an estimate for the function field analogue of V(x;q) which suggests that this range could be extended to x^{o(1)} < q< x. In this talk we will show how one can use probabilistic techniques to give evidence that V(x;q) should be asymptotic to x log q in the even wider range (log log x)^{1+o(1)} < q < x, and that this range is best possible. Mar. 11 Maksim Radziwill (Stanford University) Feb 25 Igor Shparlinski (Macquarie University) Gaps between quadratic non-residues and primitive roots in Hamming metric (Joint work Rainer Dietmann and Christian Elsholtz) We give various results about the gaps between quadratic non-residues and primitive roots modulo a prime p when the distance is measured in Hamming metric on binary expansions of integers. For example, we show that there is a primitive root $g \in [1,p-1]$ with at most $0.11003 n$ non-zero binary digits, where n is the number of binary digits of p. We also show that there is a prime quadratic non-residue $q \in [1,p-1]$ with at most $0.1172 n$ non-zero binary digits. These results are based on some recent bounds of character sums and simple combinatorial arguments. We also discuss some open problems. Feb. 11 Yu-Ru Liu (University of Waterloo) Multidimensional Vinogradov-type Estimates in Function Fields Let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$. In this talk, we will employ Wooley's new efficient congruencing method to prove certain multidimensional Vinogradov-type estimates in $\mathbb{F}_q[t]$. These results allow us to apply a variant of the circle method to obtain asymptotic formulas for a system connected to the problem about linear spaces lying on hypersurfaces defined over $\mathbb{F}_q[t]$. This is a joint work with Wentang Kuo and Xiaomei Zhao. Jan. 28 John Friedlander (U of T) Shifted squares, sifted We discuss a number of problems and results centred on the theme of applying the sieve to the sequence of integers $\{N - n^2 : n < \sqrt{N}\}$, for a given positive integer $N$. Jan. 21 Peter Cho (Fields/U of T) Probabilistic properties of number fields We study several properties of number fields. For example, let $N_K$ be the smallest prime which does not split completely in a number field $K$. Let $d_K$ be the absolute discriminant of $K$. Then we show that, with probability one $$N_K \ll (\log d_K)^m$$ for some $m>0$ when $K$ belong to certain families of number fields. Jan. 7 Xiannan Li (UIUC) The Riemann zeta function on arithmetic progressions I will talk about the distribution of the values of the zeta function on points lying in an arithmetic progression on the critical line. This research was originally motivated by questions about the primes and the linear independence conjecture. We discover some interesting correlations between such distributions along sparse discrete points and the usual distribution of values on the entire critical line. Among other applications, this allows us to prove that a positive proportion of such points are not zeros of zeta, improving a previous result of Martin and Ng. This is based on joint work with M. Radziwill. Nov. 26 Dimitris Koukoulopoulos (University of Montreal) Let $S(x,P)$ be the number of integers up to $x$ that have no prime factors from the set of primes $P\subset\{p \le x\}$. In general, a naive probabilistic heuristic suggests that $S(x,P) \approx x\cdot \prod_{pP} (1-1/p)$. Sieve methods yield good upper and lower bounds, of this size, when $P$ is a subset of the primes in $\{p \le x^{1/2-\epsilon}\}$, but they are inapplicable if $P$ contains lots of primes $>x^{1/2}$. Now, for such $P$, the size of $S(x,P)$ has been studied in only a few cases. In the case when $P= \{y \pi(x;q,a_2) > \cdots > \pi(x;q,a_r)$. We expect that this system should have arbitrarily large solutions $x$, and moreover we expect the same to be true no matter how we permute the residue classes $a_j$; if this is the case, the prime number race is called "inclusive". As it happens, the explicit formula for $\pi(x;q,a_1)$ allows us to convert prime number races into problems about sums of infinitely many random variables and the analogous inequalities among them. Rubinstein and Sarnak proved conditionally that every prime number race is inclusive; they assumed not only the generalized Riemann hypothesis but also a strong statement about the linear independence of the zeros of Dirichlet $L$-functions. On the other hand, Ford and Konyagin showed that prime number races could fail to be inclusive if the generalized Riemann hypothesis is false. I will discuss these results, as well as some work in progress with Nathan Ng where we substantially weaken the second hypothesis used by Rubinstein and Sarnak. Oct. 22 Damien Roy (University of Ottawa) Diophantine approximation with sign constraints Let a and b be real numbers such that 1, a and b are linearly independent over Q. A classical result of Dirichlet asserts that there are infinitely many triples of integers (x,y,z) such that |z+ax+by| < max(|x|,|y|,|z|)^(-2). In 1976, W. M. Schmidt asked what can be said under the restriction that x and y be positive. Upon denoting by g=1.618 the golden ratio, he proved that there are triples (x,y,z) satisfying this condition for which the product |z+ax+by|.max(|x|,|y|,|z|)^g is arbitrarily small. Although, at that time, Schmidt did not rule out the possibility that g be replaced by any number smaller than 2, Moshchevitin proved few months ago that it cannot be replaced by a number larger than 1.947. In this talk, we present a construction showing that the result of Schmidt is in fact optimal. Oct. 10 2:10 p.m. Bahen Centre, Room 6183 *please note time and location change Trevor Wooley (University of Bristol) Waring's problem, translation-invariant systems, and rational curves on hypersurfaces The new "efficient congruencing" method has very recently achieved near-optimal estimates for exponential sums associated with Waring's problem and related topics in analytic number theory. In this talk we will discuss this progress for the number of representations of a natural number as the sum of integral powers, its derivation by means of sharp estimates for the number of solutions of translation-invariant systems, and emerging applications concerning spaces of rational curves on hypersurfaces defined by diagonal equations. Oct. 1 4:00 p.m. York University N638 Ross Building *please note time and location change Cam Stewart (University of Waterloo) Arithmetic and transcendence Techniques developed for transcendental number theory have had many surprising applications in the study of purely arithmetic questions. The aim of our talk will be to discuss this phenomenon. Sept. 24 Youness Lamzouri (York) Discrepancy bounds for the distribution of the Riemann zeta function In 1930 Bohr and Jessen proved that for any $1/2 <\sigma< 1$, $\log\zeta(\sigma+it)$ has a continuous limiting distribution in the complex plane. As a consequence, it follows that the set of values of $\log\zeta(\sigma+it)$ is everywhere dense in $\mathbb{C}$. Harman and Matsumoto obtained a quantitative version of the Bohr-Jessen Theorem using Fourier analysis on a multidimensional torus. In this talk, I will present a different and more direct approach which leads to uniformdiscrepancy bounds for the distribution of $\log\zeta(\sigma+it)$ that improve the Matsumoto-Harman estimates. Sept. 17 Leo Goldmakher (U of T) www.math.toronto.edu/lgoldmak/ Lower bounds on character sums In 1932, Paley constructed an infinite family of quadratic characters with exceptionally large character sum. In this talk I will describe recent joint work with Youness Lamzouri, in which we establish an analogous result for characters of any fixed even order. Previously our results were only known under the assumption of the Generalized Riemann Hypothesis.