If prime numbers are the elementary particles of arithmetic, then the Riemann zeta function is one of the key tools for analyzing how they behave. The zeta function, although relatively easy to write down, contains multitudes: it encodes information about the distribution of primes along the number line, and is the centerpiece of unsolved problems in number theory.
Connections between the statistics of primes and physics have been made before, but now, in a paper in Physical Review Letters, Yan Fyodorov of Queen Mary, University of London, UK, and colleagues show a surprising correspondence between freezing in disordered systems, like glasses, and the peaks and valleys of the zeta function.
The energy of a disordered system is like a traveler moving around on a random landscape of hill and valleys. As the temperature is lowered, the traveler bounces from place to place but eventually settles into a local energy minimum, which marks the freezing transition in the glass. Fyodorov et al. show both analytically, and with numerical simulations, that the statistical mechanical properties of the freezing transition correlate with the statistical properties of extrema of the zeta function. Not only might this work guide the way physicists tackle important statistical physics problems, but our understanding of freezing could help mathematicians make progress in attacking some of the grand challenges of number theory. – David Voss