
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th
ANNIVERSARY
YEAR 
Number
Theory Seminar 20122013
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 zetafunction
If one could construct a model of the Riemann zetafunction that
incorporates its basic properties but has a more transparent structure,
this could provide insight into the zetafunction's behavior. I will
describe the construction of a family of functions out of finite Euler
products and any zeros the zetafunction might have to the right of
the critical line. I will then discuss how well these functions approximate
the zetafunction 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 geometryofnumbers 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 BarbanDavenportHalberstam 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 HardyLittlewood 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 nonresidues and primitive roots in Hamming
metric (Joint work Rainer Dietmann and Christian Elsholtz)
We give various results about the gaps between quadratic nonresidues
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 $g \in [1,p1]$ with at most $0.11003
n$ nonzero binary digits, where n is the number of binary digits of
p. We also show that there is a prime quadratic nonresidue $q \in [1,p1]$
with at most $0.1172 n$ nonzero 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 
YuRu Liu (University of Waterloo)
Multidimensional Vinogradovtype 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 Vinogradovtype
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 $S(x,P) \approx x\cdot
\prod_{pP} (11/p)$. 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 $P= \{y<p\le x\}$, which is known to
be the most extreme one, we have that $S(x,P)\approx x/u^u$, $u=\log
x/\log y$, much less that the expected $x/u$. Other than that not
much is known, but it is expected that, as soon as $P$ does not contains
too many big primes, the probabilistic heuristic is accurate. In this
talk, I will show that this expectation is indeed accurate: if $\sum_{y<p\le
x,\, p\notin P} 1/p \gg1$ for some $y\ge x^{O(1)}$, then $S(x;P)$
has the predicted size. This is joint work with Andrew Granville and
Kaisa Matom\"aki.

Nov. 19

JingJing Huang (U of T)
Metric Diophantine approximation on planar curves
In 1998, Kleinbock and Margulis established the fundamental BakerSprind\v{z}uk
conjecture that nondegenerate analytic manifolds are extremal. Subsequently,
the much stronger KhintchineJarn\'{i}k type theorem for nondegenerate
planar curves has been establishedthanks to Vaughan and Velani
for the convergence theory and Beresnevich, Dickinson and Velani for
the divergence theory. Though, both approaches rely on estimates on
the number of rational points with small denominators which are ``close"
to the curve, the two proofs differ quite significantly in nature.
In this talk, I will present an approach towards a unified proof of
the problem and some potential applications to the general manifolds.

Nov. 12

Adam Harper (CRM, Montreal)
A zerodensity approach to smooth numbers
A number is said to be $y$smooth if all of its prime factors are
less than $y$. Such numbers appear in many places throughout analytic
and combinatorial number theory, and much work has been done to investigate
their distribution in arithmetic progressions and in intervals.
In this talk I will try to explain the similarities and differences
between studying these problems for $y$smooth numbers and for primes.
In particular, I will explain how zerodensity results can be brought
to bear on the smooth number problems, even though there is no explicit
formula available as in the case of primes. This approach allows one
to prove results on much wider ranges of $y$ than were previously
available.

Nov. 5

Xiaoqing Li (State University of New York at Buffalo)
The L^2 restriction norm of a Maass form on GL(n+1)
In this talk, we will discuss upper and lower bounds for L^2 restriction
norms of a Maass form on GL(n+1). For certain cases, the lower bound
is unconditional and sharp. This is a joint work with ShengChi Liu
and Matt Young.

Wed. Oct. 31
*Please note nonstandard date

Greg Martin (UBC)
Inclusive prime number races
Let $\pi(x;q,a)$ denote the number of primes up to $x$ that are congruent
to $a$ (mod $q$). A "prime number race", for fixed modulus
$q$ and residue classes $a_1,\dots,a_r$, investigates the system of
inequalities $\pi(x;q,a_1) > \pi(x;q,a_2) > \cdots > \pi(x;q,a_r)$.
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, translationinvariant systems, and rational curves
on hypersurfaces
The new "efficient congruencing" method has very recently
achieved nearoptimal 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 translationinvariant
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 BohrJessen 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
MatsumotoHarman 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.

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