Harold Diamond, UIUC.
Title: Cosh-like arithmetic functions having an average value.
Florin Boca, UIUC.
Title: The correlations of Farey fractions.
Abstract:
It will be proved that all correlation measures of the sequence of Farey
fractions exist. The pair correlation function is explicitly computed and
shows an intermediate behaviour between the Poisson and the GUE correlations.
This is joint work with A. Zaharescu.
Kevin Ford, UIUC.
Title: Integers covered by systems of congruences with distinct moduli.
Abstract:
A covering system is a finite set of arithmetic progressions with distinct
moduli >1 whose union is all the integers. Erdos conjectured that there are
covering systems whose smallest moduli is arbitrarily large. We consider the
following related problem: If N is a large integer and K>1, what is the
densest union of arithmetic progressions with distinct moduli in [N,KN]? We
show that if K is a bit smaller than a power of N, then the desity of the
union of progressions cannot be much more than 1-1/K. In particular, a
covering system with distinct moduli in [N,KN] cannot exist. This is joint
work with Michael Filaseta, Sergei Konyagin, Carl Pomerance and Gang Yu.
Maria Sabitova, University of Pennsylvania.
Title: Root numbers of abelian varieties.
Abstract:
We generalize a theorem of D. Rohrlich concerning root numbers of elliptic
curves over the field of rational numbers. Our result applies to abelian
varieties over number fields. Namely, under certain conditions which
naturally extend the conditions used by D. Rohrlich, we show that the root
number W(A,\tau) associated to an abelian variety A over a number field F
and a complex finite-dimensional irreducible representation \tau of the
absolute Galois group of F with real-valued character is equal to 1. In the
case where the ground field is Q, we show that our result is consistent with
the refined version of the conjecture of Birch and Swinnerton-Dyer.
Alexandru Zaharescu, UIUC.
Title: Lehmer numbers and Lehmer points.
Abstract:
We discuss the definition of Lehmer numbers and Lehmer points, and present
some asymptotic results on the number of such points.
Lenny Fukshansky, Texas A&M.
Title: Effective structure theorems for quadratic spaces and their
isometries
Abstract:
A classical theorem of Witt states that a bilinear space can be decomposed
into an orthogonal sum of hyperbolic planes, singular, and anisotropic
components. I will discuss the existence of such a decomposition of bounded
height for a symmetric bilinear space over a number field, where all bounds
on height are explicit. I will also talk about an effective version of
Cartan-Dieudonne?? theorem on representation of an isometry of a regular
symmetric bilinear space as a product of reflections. Finally, if time
permits, I will show a special version of Siegel's Lemma for a bilinear
space, which provides a small-height orthogonal decomposition into
one-dimensional subspaces.
Emre Alkan, UIUC.
Title: Diophantine Approximation with mild divisibility constraints.
Abstract:
I will present a survey on the history of Diophantine approximation problems
using special subsets of integers and also talk about recent work on this
jointly with A. Zaharescu and G. Harman
Stephanie Treneer, UIUC.
Title: Congruences for the coefficients of weakly holomorphic
modular forms, I.
Abstract:
Recent works have used the theory of modular forms to establish linear
congruences for the partition function and for traces of singular moduli. We
show that this type of phenomena is completely general, by finding similar
congruences for the coefficients of any weakly holomorphic modular form. In
particular, we give congruences for a wide class of partitions functions and
for traces of CM values of arbitrary modular functions on certain congruence
subgroups of prime level. Tuesday's talk will consist of an introduction to
the problem, a statement of the main theorems, and a discussion of the two
applications. Thursday we will prove the main theorems.
Stephanie Treneer, UIUC.
Title: Congruences for the coefficients of weakly holomorphic
modular forms, II.
Abstract:
Recent works have used the theory of modular forms to establish linear
congruences for the partition function and for traces of singular moduli. We
show that this type of phenomena is completely general, by finding similar
congruences for the coefficients of any weakly holomorphic modular form. In
particular, we give congruences for a wide class of partitions functions and
for traces of CM values of arbitrary modular functions on certain congruence
subgroups of prime level. Tuesday's talk will consist of an introduction to
the problem, a statement of the main theorems, and a discussion of the two
applications. Thursday we will prove the main theorems.
Timothy Kilbourn, UIUC.
Title: An Extension of the Supercongruence for Apery Numbers.
Abstract:
In 1987, Beukers proved a mod p congruence between the
coefficients of a certain modular form and the Apery numbers. He
conjectured the existence of a ``supercongruence'' mod p^2; this was
proved by Ahlgren and Ono in 2000. In this talk we prove an extension of
this congruence mod p^3. The proof involves the modularity of a
Calabi-Yau threefold, the Gross-Koblitz formula, the p-adic gamma
function, and some interesting combinatorics.
Michael Drmota, Technical University, Wien.
Title: Digital expansions, exponential sums and central
limit theorems.
Abstract:
The purpose of this talk is to present recent results
on the distribution of the values of the ($q$-ary)
sum of digits function $s_q(n)$, where
we mainly focus on two problems, the joint distribution
$(s_{q_1}(n),s_{q_2}(n))$ of two different expansions
and on the distribution of the sum-of-digits function of
squares: $s_q(n^2)$.
Interestingly there are strong relations to "classical
analytic number theory". For example, one can use
Baker's theorem on linear forms (and exponential sums) to
prove that the pairs $(s_{q_1}(n),s_{q_2}(n))$, $n
Heini Halberstam, UIUC.
Title: On some very old theorems of Erdos
Paul Bateman, UIUC.
Title: The Dirichlet series for the reciprocal of the Riemann zeta
function
Abstract:
For real values of s greater than 1, we compare the partial sums of the
series $\sum \mu(n)n^{-s}$ with its sum $1/\zeta(s)$, where $\zeta$ denotes
the Riemann zeta-function. Specifically, we ask: For what values of N does
the difference
$$\sum_{n=1}^N \mu(n)n^{-s} - 1/\zeta(s) $$
have a fixed sign for all s in $(1,\infty)$. The talk is related to
my recent problem in the American Mathematical Monthly, which deals with some
simple cases.
Ben Howard, University of Chicago.
Title: The Iwasawa theoretic Gross-Zagier theorem.
Abstract:
The Gross-Zagier theorem relates the central value of the
derivative of the L-function of an elliptic curve to the Neron-Tate height
of a special point, a Heegner point, in the Mordell-Weil group. Combining
this result with work of Kolyvagin gives some of the strongest known results
in the direction of the Birch and Swinnerton-Dyer conjecture. In this
talk we will explain an Iwasawa theoretic analogue (conjectured by Mazur
and Rubin) of the Gross-Zagier theorem. The result relates towers of
Heegner points in a Z_p-extension to the derivative of a two-variable
p-adic L-function.
A. Raghuram, University of Iowa.
Title: Special values of L-functions.
Abstract:
This talk will be an introduction to Deligne's conjectures on
the special values of the symmetric power L-functions associated to a
holomorphic cuspform. The latter half of the talk will be an account of
some work in progress, which is joint work with Freydoon Shahidi, toward
the special values of the symmetric fourth power L-functions.
Bruce Berndt, UIUC.
Title: Ramanujan's 40 Identities for the Rogers-Ramanujan Functions.
Abstract:
Ramanujan's list of 40 identities was found in the Oxford University
Library by Bryan Birch and published in 1975. The list is in the
handwriting of G. N. Watson who privately held the manuscript for many years
and evidently lost Ramanujan's original list. We provide a survey of the
methods used over the years to prove the identities, with emphasis on recent
work. This is joint work with Geumlan Choi, Youn-Seo Choi, Heekyoung Hahn,
Boon Pin Yeap, Ae Ja Yee, Hamza Yesilyurt, and Jinhee Yi.
Bruce Reznick, UIUC.
Title: A computational version of a theorem of Polya.
Abstract:
In 1928, Polya proved that if p is a homogeneous polynomial
in n variables which is strictly positive on the unit simplex, then
for sufficiently large N, all the coefficients of (x1+...+xn)^N * p
are non-negative. Vicki Powers and I proved in 2001 an explicit estimate
on N, which is sharp in at least one special case. This theorem is
becoming popular in certain circles of mathematical optimization. All
proofs are elementary.
Florin Boca, UIUC.
Title: The Dirichlet series of a certain arithmetic function associated
with the continued fraction algorithm.
Ken Stolarsky, UIUC.
Title: A curious confluence of three problems in the
analytic theory of polynomials.
Abstract:
How a polynomial behaves at critical points (e.g., max-min theory) and how the
symmetry of a polynomial relates to the distribution of its zeros are two
major problem areas in the study of polynomials. A more esoteric problem is
to what extent a polynomial and some of its derivatives can all be composite
(say without having multiple zeros). Certain multivariable polynomials over
the rationals shall be displayed that seem to be remarkable (at least to me)
with respect to each of these three problem areas. They are somehow related
to Shabat polynomials.
Ben Brubaker, Stanford University.
Title: The form of Fourier coefficients of Eisenstein Series.
Abstract:
We will begin with a review of ordinary spectral Eisenstein
series on GL(2) and its Fourier coefficients and then describe a new
class of Eisenstein series associated to covers of semi-simple Lie
groups (called metaplectic groups since Weil originally worked over the
symplectic group Sp(2n)). The Fourier (or more properly Whittaker)
coefficients of these Eisenstein series have extremely rich arithmetic
information, e.g. they can contain Dirichlet L-functions or interesting
combinations of Gauss sums. Hence analytic information about the
Eisenstein series can be translated into information about the special
values of L-functions, etc. We'll present some recent examples of this
and give a combinatorial approach to dealing with these objects that
avoids much of the complication with computing the Fourier-Whittaker
coefficients directly.
Kevin Ford, UIUC.
Title: On the difference between a number and its inverse modulo n.
Abstract:
We study the distribution of the function
M(n) = max(|a-b|: 0 < a, b < n and ab=1 (mod n))
and its connection to divisor problems. In particular, we examine what can be
said (i) for infinitely many n, (ii) for infinitely many prime n, and (iii)
for almost all n.
Various faculty and graduate students in number theory. UIUC.
Title: 10 minute talks.
Abstract:
Volunteers from the regular seminar group will offer short talks in the area
of general number theory, roughly ten minutes in length.
O-Yeat Chan, UIUC.
Title: A Table of Values in Ramanujan's Lost Notebook.
Abstract:
On pages 179 and 180 of his Lost Notebook, Ramanujan lists a
table of values related to the crank generating function. In this talk, we
shall use the circle method to prove the completeness of this table, and,
as a bonus, prove 2 conjectures of Andrews and Lewis regarding cranks
modulo 3 and 4.
Andrew Booker, University of Michigan.
Title: Converse theorems and Artin's conjecture.
Abstract:
Given a finite Galois extension of the rationals and a
representation of its Galois group, Artin defined an L-function which he
conjectured to have analytic continuation to the complex plane and satisfy
a functional equation. All progress to date on this conjecture has come
through the related Langlands' program which anticipates that Artin's
L-functions agree with those associated to automorphic forms. I will show
that if Artin's conjecture is true for a given 2-dimensional
representation then so is Langlands' conjecture. The technique used in the
proof may be extended to give a Weil-type converse theorem. If time
permits, I will discuss generalizations to number fields.
Matt Boylan, UIUC.
Title: Parity of the partition function.
Abstract:
Let p(n) denote the ordinary parition function. In 1966 Subbarao conjectured
that in every arithmetic progression r (mod t) there are infinitely many
integers N = r (mod t) with p(N) even, and infinitely many integers M = r
(mod t) with p(M) odd. Using classical facts about modular forms mod 2, we
prove the conjecture for every arithmetic progression r (mod t) where t is a
power of 2. This is joint work with Ken Ono from 2001.
Ling Long, Iowa State University.
Title: Eisenstein series and representing natural numbers as sum of
integer squares.
Abstract:
In this talk we are going to address two questions.
We will give a short proof of Milne's formulae for sums of $4n^2$ and
$4n^2+4n$ integer squares using the theory of modular forms, in particular
Eisenstein series. This work was done jointly with Yifan Yang.
In the second part, we will address a question regarding the zeros of some
Eisenstein series used in the first part of the talk.
Kathrin Bringmann, University of Wisconsin.
Title: Traces of Singular Moduli on Hilbert Modular Surfaces.
Abstract:
Suppose that $p\equiv 1\pmod 4$ is a prime, and that $\Op$ is the ring of
integers of $K:=\Q(\sqrt{p})$. A classical result of Hirzebruch and Zagier
asserts that certain generating functions for the intersection numbers of
Hirzebruch-Zagier divisors on the Hilbert modular surface
$(\h\times \h)/\SL_2(\Op)$ are weight $2$ holomorphic modular forms. Using
recent work of Bruinier and Funke, we show that the generating functions of
traces of singular moduli over these intersection points are often weakly
holomorphic weight $2$ modular forms. For the singular moduli of
$J_1(z)=j(z)-744$, we explicitly determine these generating functions using
classical Weber functions, and we factorize their ``norms" as products of
Hilbert class polynomials.
Paul van Wamelen, Lousiana State University.
Title: Computing with the Jacobian of a Hyperelliptic Curve.
Abstract:
We will define the algebraic and analytic Jacobians of a hyperelliptic
curve and explain how to go from the one to the other. We will demonstrate
functionalities in the computer algebra system MAGMA for working with
these objects.