UC Irvine

We study the one-level density of zeros for several families of Dirichlet L-functions over function fields and prove results which support the connection between zeros of families of L-functions and statistics of eigenvalues of random matrices.

In Chapter 1, we introduce definitions of various objects of relevance, such as Dirichlet characters and Dirichlet L-functions over number fields, and present analogous ones over function fields Fq(t). We discuss the construction of order l Dirichlet characters over Fq[t] specifically in Section 1.3.1, for both the Kummer setting (q \equiv 1(mod l)) and the non-Kummer setting (q \not\equiv 1 (mod l). Section 1.4 dedicates to results that build connections between statistics of the Riemann zeta function and families of L-functions and random matrix theory; we also define and discuss the one-level density of zeros here in detail. Section 1.5 outlines the rest of the thesis, including statements of main theorems and a remark on the average order of non-vanishing at low-lying heights.

In Chapter 2, we study the one-level density of zeros for cubic and quartic Dirichlet L-functions over function fields in the Kummer setting. We prove the general explicit formula for order l Dirichlet L-functions in Lemma 2.1 and evaluate the main terms and error terms for each order. As a consequence of Theorems 1.1 and 1.2, we prove that the cubic and quartic families have unitary symmetry, supporting the philosophy of Katz and Sarnak.

In Chapter 3, we study the one-level density of zeros for cubic, quartic and sextic Dirichlet L-functions over function fields in the non-Kummer setting. We discuss a crucial construction of non-Kummer characters in Section 3.2, motivated by the works of Baier and Young, and David, Florea and Lalin. Similar to the Kummer setting, we evaluate the main terms and error term of the one-level density and prove that the families of cubic, quartic and sextic Dirichlet L-functions have unitary symmetry.

Appendix A somewhat extends the construction of non-Kummer characters in Section 3.2 to include characters of order equal to a Mersenne prime.