{"id":{"repo_id":"lethbridge","oai_identifier":"oai:opus.uleth.ca:10133/5441"},"canonical_url":"https://search.dev.ndltd.org/etd/lethbridge/oai:opus.uleth.ca:10133/5441","repository":{"repo_id":"lethbridge","name":"University of Lethbridge","base_url":"https://opus.uleth.ca/server/oai/request"},"display":{"title":"Perron's formula and resulting explicit bounds on sums","abstract":"By working with Perron’s formula we prove an explicit bound on ∑n≤x an/ns, where an,s ∈ C. We then prove a second explicit bound on this sum for the special case where s = 0: These bounds apply to specific sums that are involved in the Prime Number Theorem. Moreover, they are particularly useful in cases where a variant of the Riemann von-Mangoldt explicit formula is not unconditionally available. We choose to implement our bounds on M(x) =∑n≤x μ(n) and m(x) =∑n≤x μ(n/)n (with μ(n) denoting the Möbius function). This gives constants C > 0; c > 0 and x0 > 0 for which |M(x)|≤Cxexp(−c√logx) if x > x0 and a similar kind of bound for m(x): We believe that explicit bounds for M(x) and m(x) like these have never before been published.","abstract_html":"By working with Perron’s formula we prove an explicit bound on ∑n≤x an/ns, where an,s ∈ C. We then prove a second explicit bound on this sum for the special case where s = 0: These bounds apply to specific sums that are involved in the Prime Number Theorem. Moreover, they are particularly useful in cases where a variant of the Riemann von-Mangoldt explicit formula is not unconditionally available. We choose to implement our bounds on M(x) =∑n≤x μ(n) and m(x) =∑n≤x μ(n/)n (with μ(n) denoting the Möbius function). This gives constants C > 0; c > 0 and x0 > 0 for which |M(x)|≤Cxexp(−c√logx) if x > x0 and a similar kind of bound for m(x): We believe that explicit bounds for M(x) and m(x) like these have never before been published.","abstract_has_math":false,"creators":["Chalker, Kirsty A.","University of Lethbridge. Faculty of Arts and Science"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019","date_published":"2019","updated_at":"2026-07-27T20:02:45Z","subjects":["Mathematical analysis","Numbers, prime","Number theory","Numbers, complex","Sequences (Mathematics)","Perron's formula","explicit bounds","Prime Number Theorem","sums","Dissertations, Academic"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["hdl:10133/5441"],"render_values":[{"text":"hdl:10133/5441","href":null,"code":true}]}]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2019"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematical analysis","Numbers, prime","Number theory","Numbers, complex","Sequences (Mathematics)","Perron's formula","explicit bounds","Prime Number Theorem","sums","Dissertations, Academic"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["hdl:10133/5441"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.other","label":"Dc Description Other","values":["By working with Perron’s formula we prove an explicit bound on ∑n≤x an/ns, where an,s ∈ C. We then prove a second explicit bound on this sum for the special case where s = 0: These bounds apply to specific sums that are involved in the Prime Number Theorem. Moreover, they are particularly useful in cases where a variant of the Riemann von-Mangoldt explicit formula is not unconditionally available. We choose to implement our bounds on M(x) =∑n≤x μ(n) and m(x) =∑n≤x μ(n/)n (with μ(n) denoting the Möbius function). This gives constants C > 0; c > 0 and x0 > 0 for which |M(x)|≤Cxexp(−c√logx) if x > x0 and a similar kind of bound for m(x): We believe that explicit bounds for M(x) and m(x) like these have never before been published."]},{"key":"dc:title","label":"Title","values":["Perron's formula and resulting explicit bounds on sums"]}]}],"canonical_facts":{"dc:date.issued":["2019"],"dc:description.other":["By working with Perron’s formula we prove an explicit bound on ∑n≤x an/ns, where an,s ∈ C. We then prove a second explicit bound on this sum for the special case where s = 0: These bounds apply to specific sums that are involved in the Prime Number Theorem. Moreover, they are particularly useful in cases where a variant of the Riemann von-Mangoldt explicit formula is not unconditionally available. We choose to implement our bounds on M(x) =∑n≤x μ(n) and m(x) =∑n≤x μ(n/)n (with μ(n) denoting the Möbius function). This gives constants C > 0; c > 0 and x0 > 0 for which |M(x)|≤Cxexp(−c√logx) if x > x0 and a similar kind of bound for m(x): We believe that explicit bounds for M(x) and m(x) like these have never before been published."],"dc:identifier":["hdl:10133/5441"],"dc:subject":["Mathematical analysis","Numbers, prime","Number theory","Numbers, complex","Sequences (Mathematics)","Perron's formula","explicit bounds","Prime Number Theorem","sums","Dissertations, Academic"],"dc:title":["Perron's formula and resulting explicit bounds on sums"],"dc:type":["Thesis"]},"updated_at":"2026-07-27T20:02:45Z"}