**Pacific Institute for the Mathematical
Sciences **

**PIMS Number Theory CRG**

*Number
Theory Day*

Bank of Nova Scotia Room 1315

Phone: 604.291.5000

*Current
Invited Speaker List:*

Stephane Fischler (ENS)

Habiba Kadiri
(

Friedrich
Littmann (PIMS)

Nathan Ng (

Chris Rowe (PIMS)

*Schedule of Talks:*

*Accommodation Reservations and CMS Winter 2003 meeting:*

The CMS Winter 2003 Meeting will be held immediately
afterwards 6-8 December 2003 at

*Travel Reimbursement for Invited Speakers:*

- Include all original receipts and boarding passes for expenses covered.
- Fill out the form at http://www.sfu.ca/finance/travel/trclaim.xls.
- Send the completed travel claim forms and documents to:

c/o Olga German

SFU PIMS Office

EAA #120-1213

Tel: (604) 268-6655

Fax: (604) 268-6657

Email:

** Organizers: **Michael
Bennett,

Contact person:

*Abstracts and titles:*

Title: Nonvanishing
of class group L-functions at the central point.

Abstract: We consider the family
of L-functions $L(s, \chi)$ where $\chi$ is a class group character for the
imaginary-quadratic number field $\mathbb{Q}(\sqrt{-D})$. If $h$ denotes
the class number, then it is shown that for a suitable $c> 0$ at least $c h
\prod_{p \mid D}(1-1/p)$ of these functions do not
vanish at $s = 1/2$.

Stephane Fischler (ENS)

Title : Irrationaly of values related to polylogarithms

Abstract : In this joint work with Tanguy Rivoal, we obtain statements of

irrationality (or linear independence over the rationals) of numbers

connected to the values, at rational arguments, of polylogarithms. The

proof uses hypergeometric series and Pade approximation problems.

Title: Sum-free sets

Abstract: We will describe some techniques in harmonic
analysis which were introduced by I.Z. Ruzsa and the
speaker in order to count sets of integers having certain properties. In
particular, we will sketch a proof that the number of subsets of {1,...,N} which do not contain a triple satisfying x + y = z is
O(2^(N/2)), which solves a conjecture of Cameron and Erdos.

Habiba Kadiri (

Title : "An explicit zero-free region for Dirichlet L-functions."

Abstract : We establish that the Riemann zeta function never vanishes in a region to the left of the line Re s = 1 of the form: Re s > 1 - 1/(R log(|Im s|), where R is an effectively computable constant. We find R=5.70176. The method also applies in the case of Dirichlet L-functions.

Friedrich Littman (PIMS)

Title: Trigonometric inequalities related to the large sieve and
orthogonal polynomials

Abstract: Let m_{1} and m_{2} be two
discrete measures on the unit circle, both having finite support and such that
the support of m_{1} has no more elements than the support of m_{2}.
Let P(x) be a polynomial of degree at most n. This
talk deals with the problem of estimating the L^{2}(m_{1})-norm
of P(x) in terms of the L^{2}(m_{2})-norm of P(x) with a constant
that is independent of P(x). We combine the theory of orthogonal polynomials on
the unit circle with an application of Hilbert's inequality in the form of
Montgomery and Vaughan. If the support of m_{2} is equally spaced on
the unit circle and all masses have weight 1, the problem becomes a form of the
large sieve. This is work in progress.

Nathan
Ng (

`Title: Moments of derivatives of the Riemann zeta function`

Abstract: In this talk we will discuss several results
concerning the moments of the derivatives of the Riemann zeta function on the
critical line. In the cases of the second and fourth moment this problem has
been solved by Ingham and Conrey respectively. We will present a number theoretic technique
which provides information on the sixth moment.
We will also note the links between these moments and zero spacings of the zeta function.

Title: Representations of integers by certain positive
definite binary quadratic forms

Abstract: We prove a conjecture of Borwein and Choi
concerning an estimate on the square of the number of solutions of a certain
quadratic form. This is joint work with R. Murty.

Chris
Rowe (PIMS)

Title: CM-fields and Rubin's
conjecture

Abstract: In
1996, Rubin gave a generalized Stark's conjecture and discussed some
``consequences'' of his conjecture for totally real number fields. I will discuss what is known in the case of
CM-fields.