On the distribution of central values of Hecke L-functions

<p>Questions regarding the behavior of the Riemann zeta function on the critical line 1/2 + <em>it</em> can be naturally interpreted as questions regarding the family of <em>L</em>-functions over Q associated to the archimedian characters<em> ψ (k)</em> =<em> k -it</em> 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.</p> <p>Using tools from p-adic analysis which are analogues of traditional archimedean techniques, we prove the <em>q</em>-aspect analogue of Heath-Brown’s result on the twelfth power moment of the Riemann zeta function for Dirichlet <em>L</em>-functions to odd prime power moduli. In particular, our results rely on the <em>p</em>-adic method of stationary phase for sums of products and complement Nunes’ bound for smooth square-free moduli.</p> <p>We additionally prove the frequency-aspect analogue of Soundararajan’s result on extreme values of the Riemann zeta function for Hecke<em> L</em>-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<em> L</em>-functions, where the classification and extraction of diagonal terms depends on the geometry of the associated field’s complex embedding.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Open Access

Discipline thesis:degree_discipline

Mathematics

Year

2021

Author dc:creator

• White, Daniel

• Mathematics

Repository record dc:identifier

https://repository.brynmawr.edu/dissertations/230

OAI identifier oai:identifier

oai:repository.brynmawr.edu:dissertations-1236