Matt Boylan, USC.
Title: Coefficients of half-integral weight modular forms modulo l^j and a
conjecture of Newman, I.
Abstract:
In 1960, Newman conjectured that for all moduli M and all residue classes
r mod M, there are infinitely many n for which p(n), the ordinary partition
function, is congruent to r mod M. In other words, p(n) is conjectured to
"hit" every residue class infinitely often. Until recently, this conjecture
was known to be true for all r mod M for only a handful of small M. Let l
> 3 be prime and let j > 0 be an integer. In this talk we prove the
conjecture for all r mod l^j. To do so, we prove more general theorems
on the distribution of Fourier coefficients of half-integral weight modular
forms modulo prime powers. This talk will include the necessary background
on modular forms of half-integral weight.
Matt Boylan, USC.
Title: Coefficients of half-integral weight modular forms mod l^j and
a conjecture of Newman, II.
Abstract:
In 1960, Newman conjectured that for all moduli M and all residue classes
r mod M, there are infinitely many n for which p(n), the ordinary partition
function, is congruent to r mod M.
In last week's lecture, I gave some background on the conjecture, and I introduced modular forms of half-integral weight. In particular, I framed Newman's Conjecture in terms of the distribution of Fourier coefficients of half-integral weight modular forms.
Now let l > 3 be prime and j > 0 be an integer. In this talk, I will state general theorems on the distribution of Fourier coefficients of half-integral weight modular forms modulo l^j. I will then show how to use these theorems to prove Newman's Conjecture for all r mod l^j.
Matt Boylan, USC.
Title: Coefficients of half-integral weight modular forms modulo l^j
and a conjecture of Newman, III.
Abstract:
In last week's talk, I described how Newman's Conjecture for prime powers
l^j with l>3 and j>0 follows from general theorems on the distribution of
half-integral weight modular form coefficients modulo l^j.
In particular, a key ingredient in the proof of Newman's Conjecture in these cases is the following theorem (which is of interest in its own right):
If a modular form of weight lambda + 1/2 is supported on finitely many square-classes modulo l (i.e., it "looks like a linear combination of theta series"), then lambda (the integer part of the weight) is tightly restricted.
This should be viewed as the mod l analogue of the classical characteristic zero result of Vigneras which says that if a modular form of weight lambda + 1/2 is supported on finitely many square-classes in characteristic zero, then lambda = 0 or 1.
The proof of our theorem requires many of the important tools from the theory of modular forms including: twisting, Hecke operators, Shimura's correspondence, Galois representations, and filtrations. Definitions will be given.
Michael Filaseta, USC.
Title: The irreducibility of x^{2p} - x^{p} + m^{p}
Abstract:
In a paper with Florian Luca, Pante Stanica, and Rob
Underwood, we investigated the Galois groups associated with certain
polynomials. The irreducibility of the polynomials in the title, with p an
odd prime and m an integer >=2, was needed for our results, and we
obtained a few arguments for their irreducibility. In this talk, I will give
an argument based on some basic algebraic number theory. We will also
discuss in general the irreducibility of f(g(x)) where f(x) and g(x)
are polynomials with rational coefficients and f(x) is irreducible over the
rationals.
Michael Filaseta, USC.
Title: A Diophantine approach to the irreducibility of
x^(2p) - x^p + m^p
Abstract:
Last week, I gave an algebraic number theory argument showing the
irreducibility of the polynomials in the title, with p an odd prime and m an
integer > 1. This week I will discuss how a result of Y. Bilu, G. Hanrot,
P.M. Voutier on primitive prime divisors of Lucas and Lehmer numbers can be
used to attack the same polynomials. We will in fact obtain a result that is
both weaker (our result is only for primes p > 13 and p = 11) and stronger
(we consider coefficients in our trinomial of a more general form) than the
result obtained last week. This is a continuation of work with Florian Luca,
Pante Stanica, and Rob Underwood.
Bruce Berndt, University of Illinois at Urbana-Champaign.
Title: The five strangest, most fascinating, most interesting
results in Ramanujan's lost notebook (in the speaker's most
humble opinion).
Abstract:
Many of Ramanujan's results, especially from his lost notebook, are so
strange and surprising that it would seem that no one else, either in the
present or the future, would have had the foresight to discover them. Five
entries from Ramanujan's lost notebook have been chosen for presentation and
detailed discussion. Each of them is surprising. All have been proved,
except for one (as of this writing). At the conclusion of the lecture,
members of the audience will be asked to rank on supplied paper ballots
their choices from 1 to 5 as to which are the strangest, most fascinating, and
most interesting.
Furman University, Greenville, SC.
Gang Yu, USC.
Title: Sigma-Phi numbers of two prime divisors
Abstract:
For a positive integer $n$, let $f(n)$ be the sum of
$\phi(p^k)$, where $\phi$ is Euler's totient function and $p^k$ runs
over the maximal prime powers dividing $n$. $n$ is called a sigma-phi
number (or a Weis number according to Bob Vaughan) if $f(n)$ divides
$n$. Let $W_k(x)$ be the number of sigma-phi numbers up to $x$ with
precisely $k$ prime divisors. We show that $W_2(x)\ll \sqrt{x}/\log{x}$.
We hope that the audience could figure out how a similar upper bound
estimate can be obtained for $W_k(x)$, $k\geq 3$.
Gang Yu, USC.
Title: Pairs of squarefull numbers of small difference.
Abstract:
A positive integer $n$ is called squarefull (or powerful) if,
for every prime p, p|n implies that p^2|n. The number of
squarefull integers up to X is asymptotically c\sqrt{X} for some
constant c>0. In the talk I will estimate the number of pairs of
squarefull integers m,n\leq X such that |m-n| is not zero is small. This
problem was originally posed by Ogy Trifonov on the 1995 SERMON.
Holly Swisher, Ohio State University.
Title: Koike's Identities between Thompson Series and Rogers-Ramanujan
Functions}
Abstract:
At one point in his life, Ramanujan listed 40 identities involving what are
now called the Rogers-Ramanujan functions G(q) and H(q) on one side, and
products of functions of the form $Q_m = \prod_{n=1}^\infty (1-q^{mn})$ on the
other side. The identities are rather complicated and seem too difficult to
guess. Recently however, Koike devised a strategy for finding (but not
proving) these types of identities by connecting them to Thompson series. He
was able to conjecture many new Rogers-Ramanujan type identities between
G(q) and H(q), and Thompson series. Here we prove these identities.
Greg Martin, University of British Columbia.
Title: Friable values of polynomials
Abstract:
We summarize the current meager state of knowledge concerning how
often values of polynomials have only small prime factors (that is,
the values are "friable" or "smooth"). We also present some evidence,
in the form of a theorem conditional upon a suitably explicit
hypothesis on prime values of polynomials, to support a conjectured
asymptotic formula for the number of friable values of any
polynomial.