# SCIENTIFIC PROGRAMS AND ACTIVITIES

June 27, 2016

## June 27-29, 2009 Conference in Number Theory Carleton University

### INVITED SPEAKER ABSTRACTS

 Contributed Talks back to Conference page

Scott Ahlgren

Congruences for modular forms of half-integral weight

Abstract: The Fourier coefficients of modular forms of half-integral weight contain a tremendous amount of information. For example, they carry information about special values of L-functions, about elliptic curves, and about generating functions of combinatorial interest. I will describe some recent results on such modular forms (notably in the case of forms of level four) and their applications.

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George Andrews

Combinatorial aspects of integer partitions

Abstract: Within the last decade, the theory of partitions has flourished. This has primarily been an symbiotic interaction between modular forms and classical q-series. I hope in this lecture to focus on the q-series aspect of this exciting topic beginning with a recent Inventiones paper (Inv. Math., 169(2007), 37-73). We will then move on to parity questions related to B. Gordon's generalization of the Rogers-Ramanujan identities. We conclude with open questions.

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Bruce C. Berndt

Analysis in Ramanujan's Lost Notebook

Abstract: Published with Ramanujan's Lost Notebook are six partial manuscripts of Ramanujan in the handwriting of G.N. Watson. (The original manuscripts no longer exist.) We discuss some recently examined entries in three of these partial manuscripts falling under the purview of either classical analysis or classical analytic number theory. In particular, we discuss a beautiful transformation formula involving the Riemann zeta function and a certain class of integrals, which can be regarded as analogues of either theta functions, Gauss sums, or elliptic functions.

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John Friedlander

Brinkmanship in the Semi-linear Sieve

Abstract: An interesting aspect, essentially unique to the semi-linear (also known as half-dimensional) sieve, is that, when applied to a sequence possessing optimal level of distribution, it comes within a whisker of success. This opens
the possibility of gaining the goal by providing just a slight additional input from other sources that exploit special properties of the sequence. We shall exhibit some recently discovered instances of this feature found in joint work with Henryk Iwaniec.

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Steve Gonek

The First 150 Years of the Riemann Zeta Function

Abstract: This is the 150th anniversary of Riemann's pivotal 1859 paper "Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse" ("On the
number of primes less than a given magnitude"). In this survey talk for a general mathematical audience, we describe the contents of Riemann's paper, the early work it spurred in the theory of the zeta function, and how the theory developed subsequently, down to the present day. Along the way we give an overview of key features of the theory and an idea of how some of the most important results were achieved.

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A bound for the least prime ideal in the Chebotarev density problem

Abstract: A classical theorem due to Linnik gives a bound for the least prime number in an arithmetic progression. Lagarias, Montgomery and Odlyzko gave a generalization of this result to any number field. Their proof relies on some results about the distribution of the zeros of the Dedekind zeta function (zero free regions, Deuring-Heilbronn phenomenon). In this talk, I will present some new results about these zeros. As a consequence, we are able to prove an effective version of the theorem of Lagarias, Montgomery and Odlyzko.

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Ram Murty

Special values of L-series

Abstract: We will discuss the transcendental nature of special values of abelian and non-abelian L-series. The first half of the lecture will survey results in this context since the time of Dirichlet. In the second half, we will present new results with a special focus on special values of Hecke L-series of imaginary quadratic fields.

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Nathan Ng

Moments of the Riemann zeta function

Abstract: In this talk I will discuss the problem of evaluating integral and discrete moments of the Riemann zeta function. In 1918 Hardy and Littlewood introduced the integral moments in order to study the Lindelof hypothesis. In the 1980's Gonek and Hejhal independently introduced discrete moments in which the zeta function is averaged over its zeros. These discrete variants are important for various number theoretic applications such as finding simple zeros of the zeta function or finding small gaps between the zeros of the zeta function. I will discuss some of the different techniques for evaluating these moments and the challenges we face in evaluating high moments in both cases.

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Ken Ono

Generalized Borcherds products and two number theoretic applications

In his 1994 ICM lecture, Borcherds introduced a very strange construction of certain modular forms. He obtained a vast generalization of the classical fact that Delta(z), the normalized cusp form of weight 12 on SL_2(Z), is a simple infinite product. He constructed forms with Heegner divisor as infinite products whose exponents are Fourier coefficients of weight 1/2 modular forms. In joint work with Bruinier, we have constructed "generalized Borcherds products" where the infinite product exponents are coefficients of harmonic Maass forms (a generalization of classical modular forms). Here we discuss the construction and present two number theoretic
applications:
1) Parity of the partition function
2) Central values and derivatives of modular L-functions

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Kannan Soundararajan

Quantum Unique Ergodicity and Number Theory

Abstract: A fundamental problem in the area of quantum chaos is to understand the distribution of high eigenvalue eigenfunctions of the Laplacian on certain Riemannian manifolds. A particular case which is of interest to number theorists concerns hyperbolic manifolds arising as a quotient of the upper half-plane by a discrete arithmetic" subgroup of SL_2(R) (for example, SL_2(Z), and in this case the corresponding eigenfunctions are called Maass cusp forms). In this case, Rudnick and Sarnak have conjectured that the high energy eigenfunctions become equi-distributed. I will discuss some recent progress which has led to a resolution of this conjecture, and also on a holomorphic analog for classical modular forms.

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R. K. Guy and H. C. Williams

Some Interesting Divisibility Sequences

Abstract: A sequence of rational integers {A_n} is said to be a divisibility sequence if A_m | A_n whenever m | n. If the divisibility sequence {A_n} also satisfies a linear recurrence relation, it is said to be a linear divisibility sequence. Linear divisibility sequences, where the characteristic polynomial for the recurrence is irreducible and of degree larger than 2, tend to occur rather infrequently. However, consider the problem of enumerating the ways a(n) of tiling a 4xn rectangle by dominoes. It turns out that {a(n)}satisfies a linear recurrence with characteristic polynomial x^4-x^3-5x^2-x+1 with initial values 1, 5, 11, 36 for n=1,2,3,4, respectively. What is remarkable about {a(n)} is that it is also a divisibility sequence. In this talk we will describe the number theoretic properties of a much more general quartic recurrence than {a(n)}. This sequence is also a divisibility sequence and includes {a(n)} for certain values of some parameters. Also, this general sequence shares many properties with the well–known Lucas function U_n. We go on to discuss a particularly remarkable octic linear divisibility sequence which arises from counting the number of spanning trees of the graph P_4 x P_n.

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### Contributed Talks

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Yumiko Ichihara

The first moment of the value of automorphic L-functions over primitive forms on the critical line
Abstract: We are interested in the first moment of the value of automorphic L-functions L_f (1/2+it), which is a sum over primitive forms. In this talk, I will show an asymptotic formula for it, in the case that weight k is an even integer satisfying 0 < k < 12 and level is p^a, where p is a prime number. This formula yields a lower bound of the number of primitive forms which L_f(1/2+it) are not vanish.

Marie Jameson and Robert Lemke Oliver

Proof of Alder's Conjecture
Abstract: Motivated by classical identities of Euler, Schur, and Rogers and Ramanujan, H. L. Alder investigated $q_d(n)$ and $Q_d(n),$ the number of partitions of $n$ into $d$-distinct parts and into parts which are $\pm 1 \pmod{d+3},$ respectively. He conjectured that $$q_d(n) \geq Q_d(n).$$ G. E. Andrews and A. J. Yee proved the conjecture in the cases where $d = 2^s-1$ and $d \geq 32.$ We complete the proof of Alder's conjecture by determining asymptotic estimates for these partition functions with explicit error terms (correcting earlier work of G. Meinardus).

Shanta Laishram

Irreducibility of generalized Hermite-Laguerre Polynomials
Abstract: Let $a\ge 0, a_0, a_1, \cdots , a_m$ be integers. Let \begin{align*} f_a(x)=\sum^m_{j=0}\frac{a_jx^j}{(j+a)!}.
\end{align*}Schur(in 1929) proved that $f_0(x)$ with $|a_0|=|a_n|=1$ is irreducible $\forall m$. Schur's result has been generalized by many authors by using $p-$adic methods of Coleman and Filaseta. In this talk, I will give a survey of the some of these results and prove some results on the irreducibility of generalized Hermite-Laguerre Polynomials by combining $p-$adic methods with the greatest prime factor of the product of terms of an arithmetic progression.

Youness Lamzouri

Distribution of values of L-functions at the edge of the critical strip
Abstract: In 2003, Granville and Soundararajan computed large moments of the family of Dirichlet $L$-functions of quadratic characters at $s=1$ and deduced an asymptotic formula for the distribution function of the $L$-values. They also proved analogous results for the Riemann zeta function on the line Re$(s)=1$. Following their ideas, Liu, Royer and Wu studied the distribution of values of $L$-functions attached to holomorphic cusp forms in the weight aspect. In this talk we will generalize these results, namely by constructing and studying a large class of random Euler products. We then deduce information of the distribution of values of families of $L$-functions at the edge of the critical strip. Among new applications, we study families of symmetric power $L$-functions of holomorphic cusp forms in the level aspect (assuming the automorphy of these $L$-functions) at $s=1$, functions in the Selberg class in the height aspect, and quadratic twists of a fixed $GL(m)/{\Bbb Q}$ automorphic cusp form at $s=1$.

Mathieu Lemire

Extensions of the Ramanujan-Mordell formula and representations by quadratic forms
Abstract: The Mordell-Ramanujan formula concerns the number of representations of a positive integer $n$ by the form $x_{1}^{2} + x_{2}^{2} + \cdots + x_{k}^{2}.$ In a recent article, Alaca, Alaca and Williams gave an elementary proof of the Mordell-Ramanujan formula when $k$ is a multiple of $4$ by making use of some basic properties of polynomials associated with Eisenstein series. In this talk, we use similar ideas to determine extensions of the Mordell-Ramanujan formula. That is, we extend this formula to the form
$x_{1}^{2} + \cdots + x_{r}^{2} + 2 x_{r+1}^{2} + \cdots + 2 x_{r+s}^{2} + 4 x_{r+s+1}^{2} + \cdots + 4 x_{k}^{2}$ for an arbitrary positive integer $k \equiv 0 (\text{\rm{mod}} \ 4)$ and integers $r$, $s$ and $t$ satisfying $r \geq 1$, $s \mbox{(even)} \geq 0$, $t \geq 0$, $r+s+t=k$. As a result, we obtain several new explicit formulae giving the number of representations of an integer $n$ by forms satisfying these conditions when $k \in \{4,8,12,16\}$.

Matija Kazalicki

2-adic and 3-adic part of class numbers and central values of $L$-functions
Abstact: In the early 80s, Williams showed that if $\epsilon=T+U\sqrt{p}$ is a fundamental unit of the real quadratic field $\Q(\sqrt{p})$, and if $h(-p)$ is the class number of $\Q(\sqrt{-p})$ then $h(-p) \equiv T + (p-1) \pmod{16}$, where $8|h(-p)$.
In this talk we study the connection between 2-part and 3-part of class number $h(-d)$ and $h(-3d)$ and ray class groups of $\pd$ unramified outside $2$ (and $3$), when $d$ is prime or the product of two primes. We obtain certain "reflection'' theorems, and as an immediate consequence we reproduce the result of Williams (and we get a similar result in the case when $d$ is the product of two primes).
The main ingredients of the proof are certain congruences between $L_2(1,\chi_d)$ (and $L_3(1,\chi_d)$) and $h(-d)$(and $h(-3d$)) modulo powers of $2$ (and $3$), which we prove using modular forms. We also obtain similar congruences for the central values of $L$-functions associated to Ramanujan's $\Delta$-function, and relate them to the structure of $2$-adic and $3$-adic Galois representation attached to the $\Delta$-function.

Sun Kim

Göllnitz-Gordon identities and parity questionsin partitions
(with Ae Ja Yee).
Abstract: Parity has played a role in partition identities from the beginning. In his recent paper, George Andrews investigated a variety of parity questions in partition identities. At the end of the paper, he then listed 15 open problems. The purpose of this paper is to to provide answers to the first three problems from his list, which are related to the Göllnitz-Gordon identities and their generalizations.

Abdellah Sebbar

Equivariant forms, construction and structure
Abstract: The notion of equivariant forms for a modular group will be introduced. After constructing few examples, we will see how to construct algebraic and geometric structures on these forms.

Thomas Stoll

Bounds for the discrete correlation of infinite sequences on $k$ symbols and generalized Rudin-Shapiro sequences
Abstract: Pseudorandom sequences, i.e., deterministic sequences with properties reminiscent of random sequences, are a well-studied subject. In this talk, we study the discrete correlaton of infinite sequences over a finite alphabet, where we just take into account whether two symbols are identical. We show that the correlation cannot be &quot;too small&quot; in some specific sense; moreover, we construct a large class of sequences (generalizd Rudin-Shapiro sequences) which achieve the bound, provided $k$ is prime or squarefree. The proofs involve combinatorial sieving, the Lovasz local lemma and exponential sums estimates. This is joint work with E. Grant and J. Shallit.

Gary Walsh

Eclipsing Siegel's method on a family of Quartic Equations
Abstract: Shabnam Akhtari has recently refined Siegel's argument in the case of certain quartic Thue equations, providing a means to determine upper bounds for quartic equations of the form $X^2-dY^4=k$, the subject of recent work by the speaker. We show that this method can be improved substantially by way of an irrationality measure due to Yuan.

Benjamin Weiss

Galois Groups of Random P-Adic Polynomials
Abstract: The space of fixed degree polynomials with p-adic coefficients has a natural probability distribution. Each polynomial also has an associated group which is the Galois group of its splitting field. We will discuss the induced distribution on groups, and derive results for the limiting distribution as p grows. Time permitting, we will discuss a relationship to Serre's mass formulae for extensions of local fields, and prove a complementary theorem to the Chebotarev Density theorem. This work is joint with Chris Hall.

Thomas Wright

Adelic Singular Series and the Goldbach Conjecture
Abstract: The purpose of this paper is to show how adelic ideas might be used to make progress on the Goldbach Conjecture. In particular, we present a new Schwartz function which is able to keep track of the number of prime factors of an integer. We then use this, along with the Ono/Igusa adelic methods for Diophantine equations, to present an infinite sum whose evaluation would prove or disprove the veracity of the Goldbach conjecture. We also use this to compute the singular series, a quantity which, if we make the highly non-trivial assumption that the circle method of Hardy and Littlewood applies, would indicate whether there are solutions to the Goldbach equation for every even natural number. In particular, if the circle method applies, this quantity would be sufficiently large to prove Goldbach is true.

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