The focus of the seminar will be on number theory of an analytic, combinatorial, or probabilistic flavour.

Time: Fridays from 2:30 to 3:30 pmPlace: N638 Ross, York University

If you are interested in giving a talk, please contact one of the organizers at aodahl *at* yorku.ca or lamzouri * at * mathstat.yorku.ca.

Date: | April 14, 2017 |
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Speaker: | Daniel Fiorilli (Ottawa) |

Title: | Major arcs and moments of arithmetical sequences in progresssions |

Abstract: | I will discuss recent joint work with de la Bretèche on the mean and variance of arithmetical sequences in progressions. |

Date: | September 30 |
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Speaker: | Alexander (Sacha) Mangerel (Toronto) |

Title: | On Binary Correlations of Multiplicative Functions |

Abstract: |
This talk will briefly survey some of the recent results in the theory of correlations of 1-bounded, complex-valued multiplicative functions. In this context we will discuss three applications. The first two are joint work with O. Klurman. The first application is a rigidity theorem. We discuss the interesting result that, in a variety of cases, if f,g are multiplicative functions a sufficient number of whose binary correlations are identical, then f(n) = g(n)n^{it}. The second application is an effective version of a result of Frantzikinakis and Host on asymptotic formulae for Gowers norms of multiplicative functions and, more generally, for multilinear averages of k-ary correlations of multiplicative functions. The third application is to the study of the set of integers n for which a given positive multiplicative function h satisfies h(n) = h(n+1). We outline a general strategy for this problem and discuss its relationship with a conjecture of Erdos, Pomerance and Sarkozy and with the famous conjecture of Chowla on correlations of the Mobius function. |

Date: | October 7 |

Speaker: | Patrick Meisner (Concordia) |

Title: | Distribution of the number of points on curves over $F_p$ |

Abstract: | Click here for PDF |

Date: | October 14 |

Speaker: | Timothy Foo (Toronto at Mississauga) |

Title: | Prime values of cubic polynomials on average |

Abstract: | A conjecture of Bateman and Horn addresses how often an irreducible polynomial may assume prime values. This was shown to be true by Baier and Zhao for a family of quadratic polynomials on average. In joint work with Zhao, as an application of the Baier-Young large sieve for cubic Dirichlet characters, that result was extended to a family of cubic polynomials. |

Date: | October 21 |

Speaker: | Patrick Ingram (York) |

Title: | The critical height of a rational function |

Abstract: | The orbits of critical points of a rational function are of central importance in dynamics. At a conference in 2010, Silverman defined a natural (from the point of view of dynamics) measure of the complexity of these orbits, called the critical height. It is a real-valued function on the moduli space of dynamical systems of a given degree, and so it is natural to ask how it compares to other functions on that variety, such as ample Weil heights. We discuss what is and is not known about the relation between these functions. |

Date: | October 28 |

No seminar - reading week | |

Date: | November 4 |

Speaker: | J.C. Saunders (Waterloo) |

Title: | Random Fibonacci Sequences |

Abstract: | Click here for PDF |

Date: | November 11 |

Speaker: | József Vass (York) |

Title: | A Generalization of Euler's Criterion to Composite Moduli |

Abstract: | A necessary and sufficient condition is provided for the solvability of a binomial congruence with a composite modulus, circumventing its prime factorization. This is a generalization of Euler's Criterion through that of Euler's Theorem, and the concepts of order and primitive roots. Idempotent numbers play a central role in this effort. |

Date: | November 18, 2016 |

Speaker: | Omar Kihel (Brock) |

Title: | On the index of a number field |

Abstract: | Click here for PDF |

Date: | November 25 |

Speaker: | Sneha Chaubey (Illinois at Urbana-Champaign) |

Title: | Results on zeros of combinations of derivatives of completed Riemann zeta function |

Abstract: | The Riemann Hypothesis implies that the zeros of all the derivatives of the Riemann xi function \xi(s) lie on the critical line. Results on the proportion of zeros on the critical line of derivatives of \xi(s) have been investigated before, and it has been shown that the percentage of zeros of \xi(k)(s) approaches 100 percent as the order of the derivative increases. In this talk, we prove a result for combinations of derivatives of \xi(s). Although our combinations do not always have all their zeros on the critical line, we show that the proportion of zeros on the critical line tends to 1. |

Date: | January 13 |

Speaker: | David McKinnon (Waterloo) |

Title: | Diophantine approximation of algebraic points |

Abstract: | Roth's Theorem says (roughly) that a rational number p/q can't be closer than 1/q^2 to a fixed algebraic number if q is big enough. This result is rightly hailed as one of the greatest achievements in number theory in the twentieth century, but how awesome can it really be if Mike Roth (Queen's University, no relation) and I can prove a theorem that is a million times more general? Mmm? I mean honestly. |

Date: | January 20 |

Speaker: | Jack Buttcane (SUNY Buffalo) |

Title: | Number theory and analysis on GL(3) |

Abstract: |
Exponential sums, automorphic forms and L-functions on GL(2) are now standard topics in number theory, and we hope that their generalization to higher rank settings will help solve many open problems. I will discuss several open problems of particular interest on GL(3) – the symmetric square L-functions, hyper-Kloosterman sums and cubic equidistribution, and explain the relationship between these problems and the GL(3) Kloosterman sums, Poincare series and Kuznetsov formula. |

Date: | January 27, 2017 |

Speaker: | Stanley Xiao (Waterloo) |

Title: | On binary quartic forms with small Galois group |

Abstract: | In this talk, we shall give an explicit parametrization of all binary quartic forms whose $\operatorname{GL}_2(\mathbb{R})$-automorphism group contains an element which is proportional over $\mathbb{R}$ to a rational matrix, in terms of integral binary quadratic forms. The irreducible elements of this class correspond to quartic forms $F$ whose Galois group (of the splitting field of $F$) do not have elements of order 3. This is joint work with C.S. Tsang. |

Date: | February 3, 2017 |

Speaker: | Youness Lamzouri (York) |

Title: | Large fixed order character sums |

Abstract: | Click here for PDF |

Date: | February 10, 2017 |

Speaker: | Kevin Hare (Waterloo) |

Title: | Continued Logarithms |

Abstract: | Click here for PDF |

Date: | February 17, 2017 Alternate time: 11:30 AM |

Speaker: | Bruno Martin (Université du Littoral) |

Title: | On prime numbers with as many ones as zeros in their binary expansion |

Abstract: |
Drmota, Mauduit and Rivat showed in 2009 that the set $\mathcal{E}$ of prime numbers with as many ones as zeros in their binary expansion is infinite (they actually gave an asymptotic formula for $\# \mathcal{E}\cap [1, N]$ for $N\ge 2$).
We prove that for every irrational number $\beta$, the sequence
$(\beta p)_{p\in\mathcal{E}}$ is uniformly distributed modulo 1. This is a joint work with Christian Mauduit and Joel Rivat (Université d'Aix-Marseille). |

Date: | April 8, 2017 |

Speaker: | Steve Lester (CRM, Montreal) |

Title: | The distribution of sums of divisor functions in short intervals |

Abstract: | In this talk I will describe some recent work about sums of divisor functions in short intervals. In particular, I will discuss the variance of sums of divisor functions over short intervals and describe different behaviors of this variance depending upon the length of the intervals considered. Additionally, I will discuss the limiting distribution of sums of divisor functions over short intervals. Here too one sees different types of behavior depending on the lengths of the intervals. I will also raise some open questions and discuss analogues of these questions for sums of divisor functions over arithmetic progressions to large moduli. This is in part joint work with Nadav Yesha. |

Date: | April 14, 2017 |

Speaker: | Daniel Fiorilli (Ottawa) |

Title: | Major arcs and moments of arithmetical sequences in progresssions |

Abstract: | I will discuss recent joint work with de la Bretèche on the mean and variance of arithmetical sequences in progressions. |

Date: | September 28 |
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Speaker: | Youness Lamzouri (York) |

Title: | The distribution of the Riemann zeta function in the critical strip |

Abstract: | We construct a probabilistic random model for the Riemann zeta function zeta(s) inside the strip 1/2< Re(s)< 1, and investigate how well its distribution approximates that of values of zeta(s). Among our applications, we obtain the first effective error term for the number of a-points of zeta(s) (defined as the roots of zeta(s) = a) in a strip 1/2 < sigma_1 < sigma_2 < 1. We also study the joint distribution of shifts of zeta(s) and use it to improve Voronin's celebrated Universality Theorem for the zeta function. This is joint work with Steve Lester (KTH) and Maksym Radziwill (Rutgers). |

Date: | October 5 |

Speaker: | Alexander Dahl (York) |

Title: | Subconvexity for a double Dirichlet series and non-vanishing of L-functions |

Abstract: | We study a double Dirichlet series of the form $ \sum_{d} L(s,\chi_d \chi)\chi'(d)d^{-w} $, where chi and chi' are quadratic Dirichlet characters with prime conductors N and M respectively. A functional equation group isomorphic to the dihedral group of order 6 continues the function meromorphically to C^2. A convexity bound at the central point is established to be (MN)^{3/8+eps} and a subconvexity bound of (MN(M+N))^{1/6+eps} is proven. This bound is used to prove an upper bound for the smallest positive integer d such that L(1/2,\chi_{dN}) does not vanish. |

Date: | October 19 |

Speaker: | Allysa Lumley (York) |

Title: | New bounds for psi(x; q, a) |

Abstract: | Click here for PDF |

Date: | October 26 |

Speaker: | Alexey Kuznetsov (York) |

Title: | How hard is it to compute a finite sum? |

Abstract: | It should come as no surprise that evaluating a finite sum of N numbers typically requires at least N-1 additions. Some finite sums can be computed explicitly and thus require much less effort: for example, 1+2+...+N=N(N+1)/2 requires only four arithmetic operations, while 1+x+x^2+...+x^N=(x^{N+1}-1)/(x-1) needs seven of them. In this talk I will discuss a fast algorithm for evaluating a finite quadratic-exponential sum, having N terms of the form x_n=exp(ian^2+ibn). There is no known explicit formula for this finite sum, yet it can be computed in poly-logarithmic (in N) number of arithmetic operations. I will also explain how this result fits into a simple and very efficient algorithm (due to G. Hiary) for fast evaluation of the Riemann zeta function on the critical line. |

Date: | November 2 |

Speaker: | Trueman MacHenry (York) |

Title: | Uses of isobaric polynomials in number theory |

Abstract: |
Isobaric Polynomials are polynomials in k variables that are indexed by partitions of integers. They form a ring generated either by the Generalized Fibonacci Polynomials (GFP), or by Generalized Lucas Polynomials, under a convolution product, which is isomorphic to the ring of Symmetric Polynomials. The GFP and the GLP are images under this isomorphism of the well known generating sets of Complete Symmetric (Homogeneous) Polynomials and Power-sum Symmetric Polynomials. The group generated by the GFP with the convolution product, in turn, under integer evaluation, is isomorphic to the group of Multiplicative Arithmetic Functions (MF) under the Dirichlet product. And the group generated by the of Additive Arithmetic Functions (AF) is isomorphic to the group generated by the GLP under convolution product, also using the integer evaluation map. The first of these examples has been useful to solve the problem of embedding MF in its injective closure. Polya's Counting Theorem, Character Theory and Representation Theory for the Symmetric groups, Field Extension Theory are some other topics which can be usefully represented by Isobaric Polynomials, as well as applications to Cryptography. Perhaps Analytic Number theorists will also be able to find the Isobaric Theory of use. |

Date: | November 9 |

Speaker: | Tristan Freiberg (Waterloo) |

Title: | Short intervals with a given number of primes. |

Abstract: | Cram\'er's random model leads us to expect that the primes are distributed in a Poisson distribution around their mean spacing. It is conjectured that, for any given positive number $\lambda$ and nonnegative integer $m$, the proportion of $n \le x$ for which $(n,n + \lambda \log n]$ contains exactly $m$ primes is asymptotically equal to $\lambda^me^{-\lambda}/m!$ as $x \to \infty$. It is also conjectured that, for any given numbers $b > a \ge 0$, the proportion of $n \le x$ for which $d_n/\log n \in (a,b]$, where $d_n = p_{n+1} - p_n$ and $p_n$ is the $n$th smallest prime, is asymptotically equal to $\int_a^b e^{-t} dt$ as $x \to \infty$. By combining an Erd{\H o}s--Rankin type construction, which produces large gaps between consecutive primes, with the Maynard--Tao breakthrough on short gaps between primes, we are able to show that the proportion of $n \le x$ for which $\pi(n + \lambda\log n) - \pi(n) = m$ is at least $x^{1 - o(1)}$. We are also able to show that at least $12.5\%$ of nonnegative real numbers are limit points of the sequence $(d_n/\log n)$ of normalized level spacings in the primes. This includes joint work with William Banks and James Maynard. |

Date: | November 23, 2015 |

Speaker: | Alexander (Sacha) Mangerel (Toronto) |

Title: | The distribution of integers with restricted prime factors |

Abstract: | Let $E_0,\ldots,E_n$ be a partition of the set of prime numbers, and define $E_j(x) := \sum_{p \in E_j \atop p \leq x} \frac{1}{p}$. Define $\pi(x;\mbf{E},\mbf{k})$ to be the number of integers $n \leq x$ with $k_j$ prime factors in $E_j$ for each $j$. Basic probabilistic heuristics suggest that $x^{-1}\pi(x;\mbf{E},\mbf{k})$, modelled as the distribution function of a random variable, should satisfy a joint Poisson law with parameter vector $(E_0(x),\ldots,E_n(x))$, as $x \rightarrow \infty$. We will discuss the validity of these heuristics in the \emph{large deviation limit}, i.e., when $k_j \gg E_j(x)^2$ for each $j$ under very mild assumptions on the sets $E_j$. We will also discuss a generalization of a mean value theorem of Hal\’{a}sz that allows us to investigate the heuristics in the case that $E_j(x)^{\e} \ll k_j \ll E_j(x)$. |

Date: | November 30 |

Speaker: | Kevin Hare (Waterloo) |

Title: | Pisot numbers |

Abstract: | We define a Pisot number $\gamma$ as a real algebraic integer strictly greater than 1 such that all of the conjugates of $\gamma$ are strictly less than 1 in absolute value. Pisot numbers have a number of very unusual and interesting properties. In this talk we will attempt to give an overview of some of these properties. |

Date: | January 11 |

Speaker: | Adam Felix (KTH Stockholm) |

Title: | How close are the order of $a$ mod $p$ and $p-1$? |

Abstract: | Let $a \in \mathbb{Z} \setminus \{0,\pm 1\}$, and let $f_{a}(p)$ denote the order of $a$ modulo $p$, where $p \nmid a$ is prime. There are many results that suggest $p-1$ and $f_{a}(p)$ are close. For example, Artin's conjecture and Hooley's subsequent proof upon the Generalized Riemann Hypothesis. We will examine questions related to the relationship between $p-1$ and $f_{a}(p)$. |

Date: | January 18 |

Speaker: | Asif Zaman (Toronto) |

Title: | The least prime ideal and the distribution of zeros of Hecke L-functions |

Abstract: |
In 1944, Linnik famously showed unconditionally that the least prime in an arithmetic progression $a \pmod{q}$ with $(a,q) = 1$ is bounded by $q^L$ for some absolute effective constant $L > 0$, known as ?Linnik?s constant?. Many authors have computed explicit admissible values of $L$ with the current world record at $L = 5$ by Xylouris (2011), refining techniques of Heath-Brown (1992). A broad generalization of this ?least prime? problem is derived from the Chebotarev Density Theorem -- a result concerned with the distribution of prime ideals in number fields according to their splitting behaviour. We will examine this generalization, its history, some examples, and recent progress towards new explicit estimates. In particular, we will discuss related explicit results on the distribution of zeros of Hecke $L$-functions. |

Date: | January 25 |

Speaker: | Patrick Ingram (Colorado State University) |

Title: | Unlikely intersections in arithmetic dynamics |

Abstract: | Given a section of an elliptic surface over C, there will be infinitely many fibres on which that section produces a torsion point. If one has two independent sections, however, it is very unlikely that both will become torsion on the same fibres, and so one might expect that this happens only finitely often. Masser and Zannier proved a theorem about this "unlikely intersection" problem, and this talk will survey some recent work on related problems of unlikely intersections in arithmetic dynamics. |

Date: | February 1 |

Speaker: | Youness Lamzouri (York) |

Title: | Large fixed order character sums |

Abstract: | In 1932, Paley constructed an infinite family of quadratic Dirichlet characters whose character sums become exceptionally large. In this talk, I will discuss some recent work (part of it joint with Leo Goldmakher), in which we obtain analogous results for characters of any fixed order. |

Date: | February 8 |

Speaker: | Alexander Dahl (York) |

Title: | The distribution of class numbers in a special family of real quadratic fields |

Abstract: | We study the distribution of $L(1,\chi_d)$ over a special family of discriminants $d$ of real quadratic fields for which this value is $\Omega(\log \log d) $. A probabilistic random model is developed which closely approximates the distribution function in a large uniform range. Principal tools used are asymptotic formulas for the complex moments of $L(1,\chi_d)$ paired with a careful saddle point analysis. We then apply these results to investigate the number of quadratic fields in this family with a given class number, producing a result analogous to one over imaginary quadratic fields developed by K. Soundararajan. |

Date: | February 29 |

Speaker: | David Spring (Glendon College, York) |

Title: | $L$-series associated to symmetric functions mod $N$ with applications related to $\zeta(3)$, $\zeta(5)$ |

Abstract: | We develop a new theory of $L$-series based on replacing Dirichlet characters mod $N$ with symmetric functions mod $N$ in order to calculate explicitly the sums of infinite series more closely related to $\zeta(2n+1)$, specifically $\zeta(3)$, $\zeta(5)$. This generalizes the corresponding theory of sums of $L$-series associated to Dirichlet characters. |

Date: | March 7 |

Speaker: | Jack Klys (Toronto) |

Title: | Transfer principles for class groups |

Abstract: | Results relating p-rank in class groups of different number fields are called transfer principles. Examples are the Scholz reflection principle for quadratic fields and Frank Gerth's work for cubic fields. Recently Jacob Tsimerman obtained similar transfer principles as a consequence of a result on class groups of algebraic tori. We describe the problem using sheaf cohomology and build on his approach to obtain more precise versions of his transfer principles which apply to both number fields and function fields. For example, under some conditions on ramification the 2-torsion in a non-Galois quartic field and its cubic resolvent are equal up to a factor bounded independently of the discriminant of the fields. |

Date: | DEPARTMENTAL COLLOQUIUM: March 14 |

Speaker: | Jacob Tsimerman (Toronto) |

Title: | Coincidences in Number Theory |

Abstract: | Number theorists spend a lot of time deciding what the *right conjectures* are; and in certain situations, we're pretty good at it. Often this involves determining which events are *likely* to happen, and which ones aren't. For example, people were pretty sure Fermat's Last Theorem was correct long before Wiles' proof, on the principle that it would be a huge coincidence for a given x,y,z,n that x^n+y^n=z^n. On the flip side, we have extremely precise conjectures for how many twin primes there are less than 10^100, even though we can't even prove infinitely many of them exist. The recently named field of `Unlikely Intersections' is devoted to showing that certain coincidences never occur, or when they do occur its only for well-understood reasons. We'll discuss how this subject came to be, and the motivations for many of the well-known conjectures. We'll also talk about certain corners of arithmetic geometry where our understanding is painfully inadequate, and describe some of the reasons why. |

Date: | March 21 |

Speaker: | Allysa Lumley (York) |

Title: | A zero density result for the Riemann zeta function |

Abstract: | Click here for PDF |

Date: | March 28 |

Speaker: | Tatiana Hessami Pilehrood (Fields Institute) |

Title: | Multiple zeta (star) values and duality relations |

Abstract: | In recent years, there has been intensive research on the Q-linear relations between multiple zeta (star) values. In this talk, we will discuss many families of identities for multiple zeta (star) values and their q-analogs. The main result is the duality relations between multiple zeta star values and Euler sums, which are generalizations of the two-one, two-three formulas and some other multiple harmonic sum identities. Such duality relations lead to a proof of the conjecture by Ihara et al. that the Hoffman $\star$-elements $\zeta^{\star}(s_1,\dots,s_r)$ with $s_i\in\{2,3\}$ span the vector space generated by multiple zeta values over Q. Based on joint work with Khodabakhsh Hessami Pilehrood and Jianqiang Zhao. |

Date: | April 4 |

Speaker: | Alexander (Sacha) Mangerel (Toronto) |

Title: | A Refinement of a Mean Value Estimate for Complex-Valued Multiplicative Functions |

Abstract: | Given an arithmetic function $g(n)$ write $M_g(x) := \sum_{n \leq x} g(n)$. We extend and strengthen the results of a fundamental paper of Hal\'{a}sz in several ways by proving upper bounds for the ratio of $\frac{|M_g(x)|}{M_{|g|}(x)}$, for any strongly multiplicative or squarefree-supported, complex-valued function $g(n)$ under certain assumptions on the sequence $\{|g(p)|\}_p$. We further prove an asymptotic formula for this ratio in the case that $|\text{arg}(g(p))|$ is sufficiently small uniformly in $p$. In so doing, we recover a new proof of an effective lower mean value estimate for $M_{|g|}(x)$ by relating it to $\frac{x}{\log x}\prod_{p \leq x} \left(1+\frac{|g(p)|}{p}\right)$. As an application, we extend a theorem of Wirsing by finding an effective rate of convergence for the ratio $\frac{|M_g(x)|}{M_{\lambda}(x)}$, assuming this quantity converges as $x \ra \infty$, where $\lambda: \mb{N} \ra (0,\infty)$ is multiplicative, $g: \mb{N} \ra \mb{C}$ is strongly multiplicative, both are uniformly bounded on primes and $|g(n)| \leq \lambda(n)$ for every $n \in \mb{N}$. |