The Riemann zeta function ζ(s) is one of the most fundamental functions in number theory. Euler demonstrated that ζ(s) is closely connected to the prime numbers and Riemann gave proofs of the basic analytic properties of the zeta function. Values of the zeta function and its derivatives have been studied by several mathematicians. Apostol in particular gave a computable formula for the values of the derivatives of ζ(s) at s = 0. The Hurwitz zeta function ζ(s,q) is a generalization of ζ(s). We modify Apostolʼs methods to find values of the derivatives of ζ(s,q) with respect to s at s = 0. As a consequence, we obtain relations among certain important constants, the generalized Stieltjes constants. We also give numerical estimates of several values of the derivatives of ζ(s,q).
","abstract_html":"<p>The Riemann zeta function ζ(s) is one of the most fundamental functions in number theory. Euler demonstrated that ζ(s) is closely connected to the prime numbers and Riemann gave proofs of the basic analytic properties of the zeta function. Values of the zeta function and its derivatives have been studied by several mathematicians. Apostol in particular gave a computable formula for the values of the derivatives of ζ(s) at s = 0. The Hurwitz zeta function ζ(s,q) is a generalization of ζ(s). We modify Apostolʼs methods to find values of the derivatives of ζ(s,q) with respect to s at s = 0. As a consequence, we obtain relations among certain important constants, the generalized Stieltjes constants. We also give numerical estimates of several values of the derivatives of ζ(s,q).</p>","abstract_has_math":false,"creators":["Musser, Jason"],"institution":null,"degree_name":"Master of Science","degree_level":null,"degree_discipline":"Department of Mathematics and Computer Science","degree_department":null,"school":null,"contributors":["Dr. Dominic Lanphier (Director), Dr. Tilak Bhattacharya, Dr. Claus Ernst"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-08-01T07:00:00Z","date_published":"2011-08-01T07:00:00Z","updated_at":"2026-07-24T06:08:09Z","subjects":["Riemann Zeta Function","Stieltjes Constants","Series expansion","Functional equation","Mathematics","Number Theory"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.wku.edu/theses/1093","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dr. Dominic Lanphier (Director), Dr. Tilak Bhattacharya, Dr. Claus Ernst"]},{"key":"dc:creator","label":"Author","values":["Musser, Jason"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Department of Mathematics and Computer Science"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Riemann Zeta Function","Stieltjes Constants","Series expansion","Functional equation","Mathematics","Number Theory"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.wku.edu/theses/1093"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Dominic Lanphier, in the Addendum, reviews some results of Stieltjes constants and relates them to the thesis. Mark Coffey is thanked for bringing attention to this work.
"]},{"key":"dc:description.abstract","label":"Abstract","values":["The Riemann zeta function ζ(s) is one of the most fundamental functions in number theory. Euler demonstrated that ζ(s) is closely connected to the prime numbers and Riemann gave proofs of the basic analytic properties of the zeta function. Values of the zeta function and its derivatives have been studied by several mathematicians. Apostol in particular gave a computable formula for the values of the derivatives of ζ(s) at s = 0. The Hurwitz zeta function ζ(s,q) is a generalization of ζ(s). We modify Apostolʼs methods to find values of the derivatives of ζ(s,q) with respect to s at s = 0. As a consequence, we obtain relations among certain important constants, the generalized Stieltjes constants. We also give numerical estimates of several values of the derivatives of ζ(s,q).
"]},{"key":"dc:title","label":"Title","values":["Higher Derivatives of the Hurwitz Zeta Function"]}]}],"canonical_facts":{"dc:contributor":["Dr. Dominic Lanphier (Director), Dr. Tilak Bhattacharya, Dr. Claus Ernst"],"dc:creator":["Musser, Jason"],"dc:description":["Dominic Lanphier, in the Addendum, reviews some results of Stieltjes constants and relates them to the thesis. Mark Coffey is thanked for bringing attention to this work.
"],"dc:description.abstract":["The Riemann zeta function ζ(s) is one of the most fundamental functions in number theory. Euler demonstrated that ζ(s) is closely connected to the prime numbers and Riemann gave proofs of the basic analytic properties of the zeta function. Values of the zeta function and its derivatives have been studied by several mathematicians. Apostol in particular gave a computable formula for the values of the derivatives of ζ(s) at s = 0. The Hurwitz zeta function ζ(s,q) is a generalization of ζ(s). We modify Apostolʼs methods to find values of the derivatives of ζ(s,q) with respect to s at s = 0. As a consequence, we obtain relations among certain important constants, the generalized Stieltjes constants. We also give numerical estimates of several values of the derivatives of ζ(s,q).
"],"dc:identifier":["https://digitalcommons.wku.edu/theses/1093"],"dc:subject":["Riemann Zeta Function","Stieltjes Constants","Series expansion","Functional equation","Mathematics","Number Theory"],"dc:title":["Higher Derivatives of the Hurwitz Zeta Function"],"dc:type":["Thesis"],"thesis:degree_discipline":["Department of Mathematics and Computer Science"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T06:08:09Z"}