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On the Siegel zeros of quadratic fields and their applications
Razakarinoro, Faratiana Brice. 2024. On the Siegel zeros of quadratic fields and their applications. Unpublished doctoral dissertation. Stellenbosch : Stellenbosch University [online]. Available: https://scholar.sun.ac.za/handle/10019.1/131897 Thesis (PhD)--Stellenbosch University, 2024.
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Stellenbosch : Stellenbosch University
2025
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| _version_ | 1867613813384151040 |
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| access_status_str | Open Access |
| author | Razakarinoro, Faratiana Brice |
| author2 | Ralaivaosaona, Dimbinaina |
| author_browse | Ralaivaosaona, Dimbinaina Razakarinoro, Faratiana Brice |
| author_facet | Ralaivaosaona, Dimbinaina Razakarinoro, Faratiana Brice |
| author_sort | Razakarinoro, Faratiana Brice |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Razakarinoro, Faratiana Brice. 2024. On the Siegel zeros of quadratic fields and their applications. Unpublished doctoral dissertation. Stellenbosch : Stellenbosch University [online].
Available: https://scholar.sun.ac.za/handle/10019.1/131897
Thesis (PhD)--Stellenbosch University, 2024. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/131897 |
| institution | Stellenbosch University (South Africa) |
| last_indexed | 2026-06-10T12:42:06.574Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2025 |
| publishDateRange | 2025 |
| publishDateSort | 2025 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/131897 On the Siegel zeros of quadratic fields and their applications Razakarinoro, Faratiana Brice Ralaivaosaona, Dimbinaina Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Quadratic fields -- Data processing Dirichlet series Functional analysis Siegel zeros L-functions Euler's numbers -- Mathematical models Number theory UCTD Razakarinoro, Faratiana Brice. 2024. On the Siegel zeros of quadratic fields and their applications. Unpublished doctoral dissertation. Stellenbosch : Stellenbosch University [online]. Available: https://scholar.sun.ac.za/handle/10019.1/131897 Thesis (PhD)--Stellenbosch University, 2024. ENGLISH ABSTRACT: In this work, we are interested in the location of the possible exceptional real zeros of Dirichlet L-functions. These are known as the Siegel zeros or Landau–Siegel zeros. In 1975, Goldfeld and Schinzel provided absolute effective constants that give bounds on such zeros. We obtain explicit upper bounds of Siegel zeros for quadratic fields in the thesis. This is achieved by studying the structure of class groups in the case of quadratic fields together with the uses of certain known computational results. We also present two natural applications of the explicit bounds, namely an up-to-date version of the zero-free region of Dirichlet L-functions and a result on the estimate of the Euler function of imaginary quadratic fields. In addition, we revisit a result of Clark and Pollack on the size of the torsion subgroup of elliptic curves with complex multiplication over a degree d number field. In 2015, they showed that this quantity is bounded by cd log log d (c is an absolute constant and is not specified). We provide the first explicit value of c in the thesis. One of the main ingredients of the proof is the use of an uniform bound of the Euler function for imaginary quadratic fields. By adapting in details the strategy of Clark and Pollack, we establish an explicit version of their estimate by applying our previous bound of Siegel zeros, in the case of imaginary quadratic fields, with other known results. AFRIKAANSE OPSOMMING: In hierdie werk stel ons belang in die ligging van die moontlike uitsonderlike reële nulpunte van Dirichlet L-funksies. Dit staan bekend as die Siegel nul- punte of Landau–Siegel nulpunte. In 1975 het Goldfeld en Schinzel absolute effektiewe konstantes gebied wat grense op sulke nulpunte gee. Ons verkry eks- plisiete boonste grense van Siegel nulpunte vir kwadratiese velde in die tesis. Dit word bereik deur die struktuur van klasgroepe te bestudeer in die geval van kwadratiese velde saam met die gebruik van sekere bekende berekenings- resultate. Ons bied ook twee natuurlike toepassings van die eksplisiete grense aan, naamlik ’n bygewerkte weergawe van die nul-vrye gebied van Dirichlet L-funksies en ’n resultaat op die skatting van die Euler funksie van imaginêre kwadratiese velde. Daarbenewens herbesoek ons ’n resultaat van Clark en Pollack op die grootte van die torsie subgroep van elliptiese krommes met komplekse ver- menigvuldiging oor ’n graad d getalveld. In 2015 het hulle aangetoon dat hierdie getal begrens word deur cd log log d (c is ’n absolute konstante en word nie gespesifiseer nie). Ons verskaf die eerste eksplisiete waarde van c in die tesis. Een van die hoofbestanddele van die bewys is die gebruik van ’n uni- forme grens van die Euler-funksie vir imaginêre kwadratiese velde. Deur die strategie van Clark en Pollack aan te pas, vestig ons ’n eksplisiete weergawe van hul skatting deur ons vorige grens van Siegel nulpunte toe te pas, in die geval van imaginêre kwadratiese velde, met ander bekende resultate. Doctoral 2025-04-08T07:22:09Z 2025-04-08T07:22:09Z 2024-12 Thesis https://scholar.sun.ac.za/handle/10019.1/131897 Stellenbosch University viii, 92 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Quadratic fields -- Data processing Dirichlet series Functional analysis Siegel zeros L-functions Euler's numbers -- Mathematical models Number theory UCTD Razakarinoro, Faratiana Brice On the Siegel zeros of quadratic fields and their applications |
| title | On the Siegel zeros of quadratic fields and their applications |
| title_full | On the Siegel zeros of quadratic fields and their applications |
| title_fullStr | On the Siegel zeros of quadratic fields and their applications |
| title_full_unstemmed | On the Siegel zeros of quadratic fields and their applications |
| title_short | On the Siegel zeros of quadratic fields and their applications |
| title_sort | on the siegel zeros of quadratic fields and their applications |
| topic | Quadratic fields -- Data processing Dirichlet series Functional analysis Siegel zeros L-functions Euler's numbers -- Mathematical models Number theory UCTD |
| url | https://scholar.sun.ac.za/handle/10019.1/131897 |
| work_keys_str_mv | AT razakarinorofaratianabrice onthesiegelzerosofquadraticfieldsandtheirapplications |