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.
","abstract_html":"<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>","abstract_has_math":false,"creators":["White, Daniel"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Open Access","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-01-01T08:00:00Z","date_published":"2021-01-01T08:00:00Z","updated_at":"2026-07-24T01:23:53Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://repository.brynmawr.edu/dissertations/230","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["White, Daniel"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Open Access"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://repository.brynmawr.edu/dissertations/230"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["On the distribution of central values of Hecke L-functions"]}]}],"canonical_facts":{"dc:creator":["White, Daniel"],"dc:description.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.
"],"dc:identifier":["https://repository.brynmawr.edu/dissertations/230"],"dc:subject":["Mathematics"],"dc:title":["On the distribution of central values of Hecke L-functions"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Open Access"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T01:23:53Z"}