Amy Mason MMath (Oxon)

My Work

The aim of my work is to gain new insights into the properties of L-functions by using their connections with Random Matrix Theory. Random Matrix Theory was developed by physicists trying to model the quantum states of complicated atomic nuclei, while L-functions arise from almost every area of number theory. The surprising link between these two areas of mathematics (usually considered entirely separate) has yet to be fully proven, but the numerical evidence indicates a definite connection.

The simplest L-function to consider is the Riemann zeta-function, a complex function with its nontrivial zeros contained in a vertical strip, called the critical strip, on the complex plane. The Riemann hypothesis says that these zeros all lie on a single vertical line, called the critical line, so rather than worrying about where the zeros lie horizontally in the critical strip we can look at how they are distributed vertically.

In the 1970s Montgomery proved - with certain conditions - a striking similarity between the pair correlation of the zeros of the Riemann zeta-function (as the height on the critical line tends to infinity) and the pair correlation of the eigenvalues of random unitary matrices (as the size of the matrices tends to infinity). Rudnick and Sarnak showed this result can be extended to n-correlation of the Riemann zeta-function and also generalised these results to apply to all L-functions. Katz and Sarnak conjectured that if we arrange L-functions into families, we can also model statistics across these families using groups of appropriate random matrices.

Unfortunately random matrices do not directly give us all the lower-order terms for the n-correlation of zeros of the Riemann zeta-function. Luckily Conrey and Snaith showed how we can use calculations on random unitary matrices as a guide for calculating the n-correlation of the Riemann zeta-function explicitly.

I have used these ideas to calculate the n-correlation of zeroes on the critical line of L-functions from a family of elliptic curves, by looking first at how the calculations work with orthogonal random matrices. I am currently working on the repeating the calculation with a familiy of Direchlet L-functions, using random symplectic matrices.