Linear Independence of ๐-Logarithms over the Eisenstein Integers
For fixed complex ๐ with |๐|>1, the ๐-logarithm ๐ฟ๐ is the meromorphic continuation of the series โ๐>0๐ง๐/(๐๐โ1),|๐ง|<|๐|, into the whole complex plane. If ๐พ is an algebraic number field, one may ask if 1,๐ฟ๐(1),๐ฟ๐(๐) are linearly independent over ๐พ for ๐,๐โ๐พร satisfying |๐|>1,๐โ ๐,๐2,๐3,โฆ. I...
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Wiley
2010-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2010/839695 |
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| author | Peter Bundschuh Keijo Vรครคnรคnen |
| author_facet | Peter Bundschuh Keijo Vรครคnรคnen |
| author_sort | Peter Bundschuh |
| collection | DOAJ |
| description | For fixed complex ๐ with |๐|>1, the ๐-logarithm ๐ฟ๐ is the meromorphic continuation of the series โ๐>0๐ง๐/(๐๐โ1),|๐ง|<|๐|, into the whole complex plane. If ๐พ is an algebraic number field, one may ask if 1,๐ฟ๐(1),๐ฟ๐(๐) are linearly independent over ๐พ for ๐,๐โ๐พร satisfying |๐|>1,๐โ ๐,๐2,๐3,โฆ. In 2004, Tachiya showed that this is true in the Subcase ๐พ=โ, ๐โโค, ๐=โ1, and the present authors extended this result to arbitrary integer ๐ from an imaginary quadratic number field ๐พ, and provided a quantitative version. In this paper, the earlier method, in particular its arithmetical part, is further developed to answer the above question in the affirmative if ๐พ is the Eisenstein number field โโ(โ3), ๐ an integer from ๐พ, and ๐ a primitive third root of unity. Under these conditions, the linear independence holds also for 1,๐ฟ๐(๐),๐ฟ๐(๐โ1), and both results are quantitative. |
| format | Article |
| id | doaj-art-29e21e1698a942639c4cd8968cde9470 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
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| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-29e21e1698a942639c4cd8968cde94702025-08-20T03:54:36ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252010-01-01201010.1155/2010/839695839695Linear Independence of ๐-Logarithms over the Eisenstein IntegersPeter Bundschuh0Keijo Vรครคnรคnen1Mathematisches Institut, Universitรคt zu Kรถln, Weyertal 86-90, 50931 Kรถln, GermanyDepartment of Mathematics, University of Oulu, P.O. Box 3000, 90014 Oulu, FinlandFor fixed complex ๐ with |๐|>1, the ๐-logarithm ๐ฟ๐ is the meromorphic continuation of the series โ๐>0๐ง๐/(๐๐โ1),|๐ง|<|๐|, into the whole complex plane. If ๐พ is an algebraic number field, one may ask if 1,๐ฟ๐(1),๐ฟ๐(๐) are linearly independent over ๐พ for ๐,๐โ๐พร satisfying |๐|>1,๐โ ๐,๐2,๐3,โฆ. In 2004, Tachiya showed that this is true in the Subcase ๐พ=โ, ๐โโค, ๐=โ1, and the present authors extended this result to arbitrary integer ๐ from an imaginary quadratic number field ๐พ, and provided a quantitative version. In this paper, the earlier method, in particular its arithmetical part, is further developed to answer the above question in the affirmative if ๐พ is the Eisenstein number field โโ(โ3), ๐ an integer from ๐พ, and ๐ a primitive third root of unity. Under these conditions, the linear independence holds also for 1,๐ฟ๐(๐),๐ฟ๐(๐โ1), and both results are quantitative.http://dx.doi.org/10.1155/2010/839695 |
| spellingShingle | Peter Bundschuh Keijo Vรครคnรคnen Linear Independence of ๐-Logarithms over the Eisenstein Integers International Journal of Mathematics and Mathematical Sciences |
| title | Linear Independence of ๐-Logarithms over the Eisenstein Integers |
| title_full | Linear Independence of ๐-Logarithms over the Eisenstein Integers |
| title_fullStr | Linear Independence of ๐-Logarithms over the Eisenstein Integers |
| title_full_unstemmed | Linear Independence of ๐-Logarithms over the Eisenstein Integers |
| title_short | Linear Independence of ๐-Logarithms over the Eisenstein Integers |
| title_sort | linear independence of ๐ logarithms over the eisenstein integers |
| url | http://dx.doi.org/10.1155/2010/839695 |
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