Moments of the Riemann Zeta function
May 5th, 2006I managed to arrange a talk by Nina Snaith for the final event by the undergraduate maths club I helped set up this year. As at Durham, she spoke on her work connecting quantum chaos to the moments of the riemann zeta function, via random matrix theory. Turnout was good, including some postgrads and staff, and the advantage of hearing the talk here is that I benefit from any feedback or insights they have.
To calculate the moments requires knowledge of a particularly elusive coefficient gk, about which very little is known: g0 is trivially 1, Hardy and Littlewood established that g1=1 in the early part of the 20th century, Ingham proved that g2=2 in 1926. No progress was made until 1995, when Conrey and Ghosh conjectured that g3 was, of all things, 42. At a conference in Vienna, Conrey and Gonek planned to present a conjecture for g4; yet the random matrix theorists had a conjecture for all values of k. Following some frantic checking at a blackboard, it was confirmed that the two conjectures agreed for g4=24024: and thus that quantum physics really could offer predictions about number theory!
One of my lecturers plugged the coefficients 1,1,2,42,24024 into the OEIS, and just one result comes up, sequence A039622. Curiously, this is about Young diagrams, which are linked to irreducible representations of the symmetric group. Young Tableau themselves describe a fairly elegant number theory puzzle. I’ve been studying representation theory this semester, and I’m always amazed by how different strands of mathematics can pull together like this- from quantum theory to distribution of primes to symmetry, somehow they’re all woven together.
For an overview of the current status of work on the Riemann Hypothesis, including Random Matrix Theory, see this article by Conrey. Marcus Du Sautoy’s the music of the primes also mentions Nina’s work, and Riemann’s own study of problems in Physics alongside the zeta function.