One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means
Available proofs of the result of the type "at least one of the odd zeta values ζ(5), ζ(7),…, ζ(s) is irrational" make use of the saddle-point method or of linear independence criteria, or both. These two remarkable techniques are, however, counted as highly non-elementary, therefore leavi...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2018 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2018
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/209436 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means / W. Zudilin // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 10 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862533140366491648 |
|---|---|
| author | Zudilin W. |
| author_facet | Zudilin W. |
| citation_txt | One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means / W. Zudilin // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 10 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Available proofs of the result of the type "at least one of the odd zeta values ζ(5), ζ(7),…, ζ(s) is irrational" make use of the saddle-point method or of linear independence criteria, or both. These two remarkable techniques are, however, counted as highly non-elementary, therefore leaving the partial irrationality result inaccessible to the general mathematics audience in all its glory. Here we modify the original construction of linear forms in odd zeta values to produce, for the first time, an elementary proof of such a result — a proof whose technical ingredients are limited to the prime number theorem and Stirling's approximation formula for the factorial.
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| first_indexed | 2025-11-24T07:13:48Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-209436 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-24T07:13:48Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Zudilin W. 2025-11-21T18:50:20Z 2018 One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means / W. Zudilin // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 10 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 11J72; 11M06; 33C20 arXiv: 1801.09895 https://nasplib.isofts.kiev.ua/handle/123456789/209436 https://doi.org/10.3842/SIGMA.2018.028 Available proofs of the result of the type "at least one of the odd zeta values ζ(5), ζ(7),…, ζ(s) is irrational" make use of the saddle-point method or of linear independence criteria, or both. These two remarkable techniques are, however, counted as highly non-elementary, therefore leaving the partial irrationality result inaccessible to the general mathematics audience in all its glory. Here we modify the original construction of linear forms in odd zeta values to produce, for the first time, an elementary proof of such a result — a proof whose technical ingredients are limited to the prime number theorem and Stirling's approximation formula for the factorial. I thank Stéphane Fischler, Tanguy Rivoal, Johannes Sprang, and the anonymous referees for their feedback on the manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means Article published earlier |
| spellingShingle | One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means Zudilin W. |
| title | One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means |
| title_full | One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means |
| title_fullStr | One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means |
| title_full_unstemmed | One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means |
| title_short | One of the Odd Zeta Values from ζ(5) to ζ(25) Is Irrational. By Elementary Means |
| title_sort | one of the odd zeta values from ζ(5) to ζ(25) is irrational. by elementary means |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209436 |
| work_keys_str_mv | AT zudilinw oneoftheoddzetavaluesfromζ5toζ25isirrationalbyelementarymeans |