Hecke's inverse cotangent numbers
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915179703304192 |
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| author | Girstmair, Kurt |
| author_facet | Girstmair, Kurt |
| contents | In connection with Eisenstein series for the principal congruence subgroup $Γ(n)$, Hecke introduced certain numbers, of which he said that they are rational and cumbersome to calculate. We show, however, that these numbers are (essentially) generators of the $n$th cyclotomic field or its maximal real subfield. They arise from the well investigated {\em cotangent numbers} by matrix inversion, which is why we call them {\em inverse cotangent numbers}. We describe them as linear combinations of roots of unity with rational coefficients in a fairly closed form, provided that $n$ is square-free. We also exhibit a formula for these numbers in terms of generalized Bernoulli numbers and Gauss sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_01349 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hecke's inverse cotangent numbers Girstmair, Kurt Number Theory 11R18 In connection with Eisenstein series for the principal congruence subgroup $Γ(n)$, Hecke introduced certain numbers, of which he said that they are rational and cumbersome to calculate. We show, however, that these numbers are (essentially) generators of the $n$th cyclotomic field or its maximal real subfield. They arise from the well investigated {\em cotangent numbers} by matrix inversion, which is why we call them {\em inverse cotangent numbers}. We describe them as linear combinations of roots of unity with rational coefficients in a fairly closed form, provided that $n$ is square-free. We also exhibit a formula for these numbers in terms of generalized Bernoulli numbers and Gauss sums. |
| title | Hecke's inverse cotangent numbers |
| topic | Number Theory 11R18 |
| url | https://arxiv.org/abs/2503.01349 |