Degree Date
2021
Degree
Doctor of Philosophy (PhD)
Department
Mathematics
Abstract
Questions regarding the behavior of the Riemann zeta function on the critical line 1/2 + it can be naturally interpreted as questions regarding the family of L-functions over Q associated to the archimedian characters ψ (k) = k -it at the center point 1/2. There are many families of characters besides those strictly of archimedean-type, especially as one expands their scope to proper finite extensions of Q. Consideration of these Hecke characters leads immediately to analogous questions concerning their associated L-functions.
Using tools from p-adic analysis which are analogues of traditional archimedean techniques, we prove the q-aspect analogue of Heath-Brown’s result on the twelfth power moment of the Riemann zeta function for Dirichlet L-functions to odd prime power moduli. In particular, our results rely on the p-adic method of stationary phase for sums of products and complement Nunes’ bound for smooth square-free moduli.
We additionally prove the frequency-aspect analogue of Soundararajan’s result on extreme values of the Riemann zeta function for Hecke L-functions to angular characters over imaginary quadratic number fields. This result relies on the resonance method, which is applied for the first time to this family of L-functions, where the classification and extraction of diagonal terms depends on the geometry of the associated field’s complex embedding.
Citation
White, D. 2021. "On the distribution of central values of Hecke L-functions.” PhD Diss., Bryn Mawr College.