Transcendence of $L(1,\chi _s)/\Pi $ in positive characteristic. A simple automata-style proof
For the field of formal Laurent series over a finite field, L. Carlitz defined $\Pi $, an analog of the real number $\pi $, and D. Goss defined $L(s,\chi )$, analogs of Dirichlet $L$-functions. G. Damamme proved in 1999 the transcendence of $L(1,\chi _s)/\Pi $ via a criterion of de Mathan. Then Y. H...
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Académie des sciences
2023-07-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.493/ |
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| author | Liu, Si-Han Yao, Jia-Yan |
| author_facet | Liu, Si-Han Yao, Jia-Yan |
| author_sort | Liu, Si-Han |
| collection | DOAJ |
| description | For the field of formal Laurent series over a finite field, L. Carlitz defined $\Pi $, an analog of the real number $\pi $, and D. Goss defined $L(s,\chi )$, analogs of Dirichlet $L$-functions. G. Damamme proved in 1999 the transcendence of $L(1,\chi _s)/\Pi $ via a criterion of de Mathan. Then Y. Hu gave in 2018 an automata-style proof of the above result. In this work, we present another and much simpler automata-style proof. |
| format | Article |
| id | doaj-art-2cfd19adb78e4d4f97e64943f13da7e8 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-07-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-2cfd19adb78e4d4f97e64943f13da7e82025-02-07T11:08:08ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-07-01361G595395710.5802/crmath.49310.5802/crmath.493Transcendence of $L(1,\chi _s)/\Pi $ in positive characteristic. A simple automata-style proofLiu, Si-Han0Yao, Jia-Yan1Department of Mathematics, Tsinghua University, Beijing 100084, P. R. ChinaDepartment of Mathematics, Tsinghua University, Beijing 100084, P. R. ChinaFor the field of formal Laurent series over a finite field, L. Carlitz defined $\Pi $, an analog of the real number $\pi $, and D. Goss defined $L(s,\chi )$, analogs of Dirichlet $L$-functions. G. Damamme proved in 1999 the transcendence of $L(1,\chi _s)/\Pi $ via a criterion of de Mathan. Then Y. Hu gave in 2018 an automata-style proof of the above result. In this work, we present another and much simpler automata-style proof.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.493/ |
| spellingShingle | Liu, Si-Han Yao, Jia-Yan Transcendence of $L(1,\chi _s)/\Pi $ in positive characteristic. A simple automata-style proof Comptes Rendus. Mathématique |
| title | Transcendence of $L(1,\chi _s)/\Pi $ in positive characteristic. A simple automata-style proof |
| title_full | Transcendence of $L(1,\chi _s)/\Pi $ in positive characteristic. A simple automata-style proof |
| title_fullStr | Transcendence of $L(1,\chi _s)/\Pi $ in positive characteristic. A simple automata-style proof |
| title_full_unstemmed | Transcendence of $L(1,\chi _s)/\Pi $ in positive characteristic. A simple automata-style proof |
| title_short | Transcendence of $L(1,\chi _s)/\Pi $ in positive characteristic. A simple automata-style proof |
| title_sort | transcendence of l 1 chi s pi in positive characteristic a simple automata style proof |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.493/ |
| work_keys_str_mv | AT liusihan transcendenceofl1chispiinpositivecharacteristicasimpleautomatastyleproof AT yaojiayan transcendenceofl1chispiinpositivecharacteristicasimpleautomatastyleproof |