# RHD

18 November 2009

The University of Auckland, City Campus

Lecture Room SLT1, Building 303, 38 Princes Street, Auckland CBD

The Riemann hypothesis was first posed in 1859, in a landmark paper on the zeta function. 150 years later, it is one of Clay Mathematics Institute’s “Millennium Problems” with a one million dollar reward for its solution.

Celebratory events are being held worldwide on November 18, 2009 — the Riemann Hypothesis Day.

Join us for an afternoon of lectures around the topic of the Riemann hypothesis and related mathematics.

The talks will be of a survey nature and accessible to all mathematicians.

Programme
• 1:00 — 1:40 Steven Galbraith (Auckland)
Welcome.
Introduction to the Riemann hypothesis and analogues for function fields
Abstract: The talk will define the Riemann zeta function, introduce the Riemann hypothesis and give some historical remarks. I will then describe the analogue of the zeta function for polynomials modulo a prime (i.e., function fields) and the proof of the Riemann hypothesis in this setting.
The holomorphic flow of the Riemann Zeta function
Abstract: (This is joint work with A. Ross Barnett.)
In 1914 Hardy proved that there were an infinite number of zeros of the Riemann zeta function on the critical line. This method was developed further by Hardy and Littlewood, Levinson, Selberg and lastly Conrey, who in 1989 showed that more than 2/5 of the zeros of the zeta function are on the critical line. However in 1934 Titchmarsh used a completely different idea, based on a good estimate for the zeta function, and using averaging over the points where zeta is real (the Gram points), to prove Hardy's theorem, and to show that the average of the zeta function over these points is 2. In this lecture we will show how the holomorphic flows determined by the zeta function and its symetric counterpart and can be used to extend Titchmarsh's average result to the whole of the right had side of the critical strip. In addition the the symmetric flow reveals that each simple zero on the critical line has an associated period, and that the logs of the periods obey a linear law".
• 2:40 — 3:20 Coffee
• 3:20 — 4:00 Shaun Cooper (Massey Albany)
On the construction of modular forms
Abstract: Riemann noted that the functional equation for the zeta function can be deduced from the behavior of the Jacobian theta function under the transformation $T:\tau \rightarrow -1/\tau$. The theta function turns out to be trivially invariant under the transformation $U:\tau\rightarrow \tau+2$, and the transformations $T$ and $U$ generate a subgroup of SL(2,Z).
In this talk I will describe some methods for constructing functions that transform nicely under certain subgroups of SL(2,Z).
• 4:00 — 5:00 Cristian Calude (Auckland), Elena Calude (Massey), Michael Dinneen (Auckland)
The complexity of the Riemann Hypothesis

Abstract: Arguably the most famous open problem problem in mathematics, the Riemann hypothesis is number 8 of Hilbert's problems and number 1 of Smale's problems. It is only natural to ask the question: how difficult is the Riemann hypothesis?
In this talk we will present a computational method for evaluating the complexity of a mathematical problem. The method is uniform and allows comparisons between problems in different fields; it cannot be applied to any problem, but works for a large class of problems including the Riemann hypothesis. We will evaluate the complexity of the Riemann hypothesis by using a suitable diophantine representation. We will show that the Riemann hypothesis has roughly the same complexity as the four color theorem, three times higher than the complexity of Fermat's last theorem and six times higher than Goldbach's conjecture.

Details: Parking for visitors is available in the Owen Glenn building at \$4 per hour. Maps of the campus are here. If you have any questions then please contact the organiser (email address below).

Organiser: Steven Galbraith

Image: Riemann Zeta function in the complex plane, generated with Mathematica by Jan Hofmann (source: wikipedia)