Number Theory and Combinatorics Seminar

Fall 2007, Spring 2008, Fall 2008, Spring 2009, Fall 2009, Spring 2010, Fall 2010

Wednesday,
September 29, 2010

Room: A580

Time: 2:00-2:50 PM

Speaker: Tim Trudgian (University of Lethbridge)

Title: Gram's Law and the zeroes of the Riemann zeta-function (I)

Abstract: That all the complex zeroes of the zeta-function lie on the critical line is the Riemann Hypothesis. Regrettably, the margin for this abstract is too narrow to write down a proof. The first 15 zeroes on the critical line were found by Gram in 1903. As a general rule of thumb, he proposed what is now called 'Gram's Law', a phenomenon by the use of which one can locate zeroes on the critical line. All subsequent searches for zeroes on the critical line use this method, in some form or another. In this seminar I shall summarise the theory one requires to state Gram's Law, as well as providing motivation for why it might be true significantly often.

Wednesday, October 6, 2010

Room: A580

Time: 2:00-2:50 PM

Speaker: Tim Trudgian (University of Lethbridge)

Title: Gram's Law and the zeroes of the Riemann zeta-function (II)

Abstract: With Gram's Law clearly defined, I shall develop the necessary theory to see whether it fails or holds in a positive proportion of cases. Several versions of Gram's Law hold and fail at this frequency, and this was my doctoral research at Oxford. Motivation from random-matrix theory will perhaps indicate the true rate of success and failure of Gram's Law.

Wednesday, October 27, 2010

Room: A580

Time: 2:00-2:50 PM

Speaker: Nathan Ng (University of Lethbridge)

Title: Nonzero values of Dirichlet L-functions in vertical arithmetic progressions

Abstract: An open question in analytic number theory is: How do the zeros of a Dirichlet L-function behave in the critical strip? One might wonder whether the zeros bunch up or are they very well-spaced. In particular, is it possible for the zeros to lie in an arithmetic progression?

In joint work with Greg Martin, we show that many terms of an arithmetic progression are not zeros of a fixed Dirichlet L-function.

Wednesday, November 24, 2010

Room: A580

Time: 2:00-2:50 PM

Speaker: Amir Akbary (University of Lethbridge)

Title: Uniform distribution of zeros of the Riemann zeta function

Abstract: We explain the Weyl criterion for the uniform distribution mod 1 of a sequence of real numbers. As an application of this criterion we describe a theorem of Hlawka on the uniform distribution mod 1 of the imaginary parts of the zeros of the Riemann zeta function.

Wednesday, December 8, 2010

Room: E575

Time: 2:00-2:50 PM

Speaker: Dave Morris (University of Lethbridge)

Title: Reconstruction from Vertex-Switching

Abstract: The Reconstruction Conjecture is a famous unsolved problem in graph theory. We will discuss a related problem that was partially solved by Richard Stanley in 1985. He used the Radon Transform, which is a technique that originated in analysis, and is the mathematical basis of modern CAT scans (used in medical diagnostics).

Time: 2:00-2:50 PM

Speaker: Tim Trudgian (University of Lethbridge)

Title: Gram's Law and the zeroes of the Riemann zeta-function (I)

Abstract: That all the complex zeroes of the zeta-function lie on the critical line is the Riemann Hypothesis. Regrettably, the margin for this abstract is too narrow to write down a proof. The first 15 zeroes on the critical line were found by Gram in 1903. As a general rule of thumb, he proposed what is now called 'Gram's Law', a phenomenon by the use of which one can locate zeroes on the critical line. All subsequent searches for zeroes on the critical line use this method, in some form or another. In this seminar I shall summarise the theory one requires to state Gram's Law, as well as providing motivation for why it might be true significantly often.

Wednesday, October 6, 2010

Room: A580

Time: 2:00-2:50 PM

Speaker: Tim Trudgian (University of Lethbridge)

Title: Gram's Law and the zeroes of the Riemann zeta-function (II)

Abstract: With Gram's Law clearly defined, I shall develop the necessary theory to see whether it fails or holds in a positive proportion of cases. Several versions of Gram's Law hold and fail at this frequency, and this was my doctoral research at Oxford. Motivation from random-matrix theory will perhaps indicate the true rate of success and failure of Gram's Law.

Wednesday, October 27, 2010

Room: A580

Time: 2:00-2:50 PM

Speaker: Nathan Ng (University of Lethbridge)

Title: Nonzero values of Dirichlet L-functions in vertical arithmetic progressions

Abstract: An open question in analytic number theory is: How do the zeros of a Dirichlet L-function behave in the critical strip? One might wonder whether the zeros bunch up or are they very well-spaced. In particular, is it possible for the zeros to lie in an arithmetic progression?

In joint work with Greg Martin, we show that many terms of an arithmetic progression are not zeros of a fixed Dirichlet L-function.

Wednesday, November 24, 2010

Room: A580

Time: 2:00-2:50 PM

Speaker: Amir Akbary (University of Lethbridge)

Title: Uniform distribution of zeros of the Riemann zeta function

Abstract: We explain the Weyl criterion for the uniform distribution mod 1 of a sequence of real numbers. As an application of this criterion we describe a theorem of Hlawka on the uniform distribution mod 1 of the imaginary parts of the zeros of the Riemann zeta function.

Wednesday, December 8, 2010

Room: E575

Time: 2:00-2:50 PM

Speaker: Dave Morris (University of Lethbridge)

Title: Reconstruction from Vertex-Switching

Abstract: The Reconstruction Conjecture is a famous unsolved problem in graph theory. We will discuss a related problem that was partially solved by Richard Stanley in 1985. He used the Radon Transform, which is a technique that originated in analysis, and is the mathematical basis of modern CAT scans (used in medical diagnostics).