From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond

Abstract

The discovery of connections between the distribution of energy levels of heavy nuclei and spacings between prime numbers has been one of the most surprising and fruitful observations in the twentieth century. The connection between the two areas was first observed through Montgomery's work on the pair correlation of zeros of the Riemann zeta function. As its generalizations and consequences have motivated much of the following work, and to this day remains one of the most important outstanding conjectures in the field, it occupies a central role in our discussion below. We describe some of the many techniques and results from the past sixty years, especially the important roles played by numerical and experimental investigations, that led to the discovery of the connections and progress towards understanding the behaviors. In our survey of these two areas, we describe the common mathematics that explains the remarkable universality. We conclude with some thoughts on what might lie ahead in the pair correlation of zeros of the zeta function, and other similar quantities.

Biases in Moments of the Dirichlet Coefficients in One-Parameter Families of Elliptic Curves

“…Hasse's theorem is very similar to the Central Limit Theorem, which itself is an example of the philosophy of square-root cancelation: if we have N objects of size 1 with random signs, then frequently the sum is of size 0 with fluctuations on the order of √ N . 3 In our setting, we expect half of the time x 3 +A(t)x+B(t) p equals 1, and the other half time it is -1. As we have p terms of size 1 with random signs, the results should be of size √ p, which is Hasse's theorem.…”

“…Before stating it, we first describe some of the problems and methods of modern number theory to motivate both why we care about this conjecture, as well as the main topic of this paper. For more on this story, see [3,9].…”