Pacific Institute for the Mathematical Sciences

PIMS Number Theory CRG

Number Theory Day

December 5, 2003 10am – 5pm

Simon Fraser University

Bank of Nova Scotia Room 1315

Harbour Center

515 West Hastings Street
Vancouver, B.C.
Canada. V6B 5K3
Phone: 604.291.5000

Current Invited Speaker List:

Valentin Blomer (University of Toronto)

Stephane Fischler (ENS)

Ben Green (PIMS)

Habiba Kadiri (University of Montreal)

Friedrich Littmann (PIMS)

Nathan Ng (University of Montreal)

Robert Osburn (Queen's University)

Chris Rowe (PIMS)

Schedule of Talks:

10:00-10:40: Ben Green

10:45-11:25: Valentin Blomer

11:30-12:10: Robert Osburn

12:15-13:00: Friedrich Littmann

13:00-14:00: Lunch

14:00-14:40: Nathan Ng

14:45-15:25: Habiba Kadiri

15:30-16:10: Stephane Fischler

16:15-17:00: Chris Rowe

Accommodation Reservations and CMS Winter 2003 meeting:

The CMS Winter 2003 Meeting will be held immediately afterwards 6-8 December 2003 at Harbour Center. Some events at this meeting include a special session in Number Theory (Sat/Sun) and a short course on Cryptography (Sat). Please refer to the links on CMS Winter 2003 meeting website www.cms.math.ca/Events/winter03, in particular www.cms.math.ca/Events/winter03/hotelse.pdf for accommodation information.

Travel Reimbursement for Invited Speakers:

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

c/o Olga German

SFU PIMS Office

8888 University Drive

EAA #120-1213

Simon Fraser University

Burnaby, B.C.

Canada V5A 1S6

Tel: (604) 268-6655

Fax: (604) 268-6657

Email: ogerman@pims.math.ca

Organizers: Michael Bennett, Peter Borwein, David Boyd, Imin Chen, Stephen Choi

Contact person: Imin Chen (ichen@math.sfu.ca)

Abstracts and titles:

Valentin Blomer (University of Toronto)

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.

Ben Green (PIMS)

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 (University of Montreal)

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 m₁ and m₂ be two discrete measures on the unit circle, both having finite support and such that the support of m₁ has no more elements than the support of m₂. Let P(x) be a polynomial of degree at most n. This talk deals with the problem of estimating the L²(m₁)-norm of P(x) in terms of the L²(m₂)-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 m₂ 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 (University of Montreal)

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.

Robert Osburn (Queen’s University)

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.