#20 Quantum mechanics, chaos,
and the music of the primes
by Michael Berry (Bristol), September 21st 2005,
2-4pm in RSLT 15
The Riemann hypothesis can be
interpreted as declaring that the prime numbers contain 'music', whose
component frequencies are the Riemann zeros. The question "Frequencies of
what?" leads to tantalizing connections with the energy levels of quantum
systems whose corresponding classical motion is chaotic. At the level of
statistics, predictions for the Riemann zeros based on semiclassical quantum
asymptotics (with primes as periods of classical trajectories) have reached a
high degree of accuracy and refinement. For the zeros themselves, the
Riemann-Siegel formula and its improvements lead to new ways of calculating
quantum levels.