We use the method of Radziwill and Soundararajan to prove Selberg’s central limit theorem for the real part of the logarithm of the Riemann zeta function on the critical line in the multivariate case. This gives an alternate proof of a result of Bourgade. An upshot of the method is to determine a rate of convergence in the sense of the Dudley distance. This is the same rate Selberg claims using the Kolmogorov distance. We also achieve the same rate of convergence in the case of Dirichlet L-functions. Assuming the Riemann hypothesis, we improve the rate of convergence by using an approximation for the logarithm of zeta given by Selberg.
","abstract_html":"<p>We use the method of Radziwill and Soundararajan to prove Selberg’s central limit theorem for the real part of the logarithm of the Riemann zeta function on the critical line in the multivariate case. This gives an alternate proof of a result of Bourgade. An upshot of the method is to determine a rate of convergence in the sense of the Dudley distance. This is the same rate Selberg claims using the Kolmogorov distance. We also achieve the same rate of convergence in the case of Dirichlet L-functions. Assuming the Riemann hypothesis, we improve the rate of convergence by using an approximation for the logarithm of zeta given by Selberg.</p>","abstract_has_math":false,"creators":["Roberts, Asher"],"institution":"The Graduate School and University Center of The City University of New York","degree_name":"Doctor of Philosophy","degree_level":"Doctoral","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":["Louis-Pierre Arguin"],"committee_chairs":[],"committee_members":["Alexander Gamburd","Matthew Junge"],"year":2024,"date_issued":"2024-09-01T07:00:00Z","date_published":"2024-09-01T07:00:00Z","updated_at":"2026-07-24T01:59:05Z","subjects":["Number Theory","Other Mathematics","Probability","riemann zeta function","central limit theorem"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://academicworks.cuny.edu/gc_etds/5913","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Louis-Pierre Arguin"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Alexander Gamburd","Matthew Junge"]},{"key":"dc:creator","label":"Author","values":["Roberts, Asher"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2024-05-10T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["The Graduate School and University Center of The City University of New York"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Number Theory","Other Mathematics","Probability","riemann zeta function","central limit theorem"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://academicworks.cuny.edu/gc_etds/5913"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We use the method of Radziwill and Soundararajan to prove Selberg’s central limit theorem for the real part of the logarithm of the Riemann zeta function on the critical line in the multivariate case. This gives an alternate proof of a result of Bourgade. An upshot of the method is to determine a rate of convergence in the sense of the Dudley distance. This is the same rate Selberg claims using the Kolmogorov distance. We also achieve the same rate of convergence in the case of Dirichlet L-functions. Assuming the Riemann hypothesis, we improve the rate of convergence by using an approximation for the logarithm of zeta given by Selberg.
"]},{"key":"dc:title","label":"Title","values":["Limit Theorems for L-functions in Analytic Number Theory"]}]}],"canonical_facts":{"dc:contributor.advisor":["Louis-Pierre Arguin"],"dc:contributor.committeemember":["Alexander Gamburd","Matthew Junge"],"dc:creator":["Roberts, Asher"],"dc:date.available":["2024-05-10T07:00:00Z"],"dc:description.abstract":["We use the method of Radziwill and Soundararajan to prove Selberg’s central limit theorem for the real part of the logarithm of the Riemann zeta function on the critical line in the multivariate case. This gives an alternate proof of a result of Bourgade. An upshot of the method is to determine a rate of convergence in the sense of the Dudley distance. This is the same rate Selberg claims using the Kolmogorov distance. We also achieve the same rate of convergence in the case of Dirichlet L-functions. Assuming the Riemann hypothesis, we improve the rate of convergence by using an approximation for the logarithm of zeta given by Selberg.
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