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Zbl 0717.11048
Girstmair, Kurt
Dirichlet convolution of cotangent numbers and relative class number formulas. (English)
[J] Monatsh. Math. 110, No.3-4, 231-256 (1990). ISSN 0026-9255; ISSN 1436-5081/e

Let n be the conductor of an absolutely abelian number field K. The ``cotangent numbers'' $i\cdot \cot (\pi k/n)$, $(k,n)=1$, belong to the n-th cyclotomic field. Their K-traces generate an additive subgroup $S\sb K$ of the ring ${\cal O}\sb K$ of integers of K. Previously [J. Number Theory 32, 100-110 (1989; Zbl 0675.12002)] we have shown that the group index of $S\sb K$ in ${\cal O}\sb K\cap i\cdot {\bbfR}$ equals $h\sp-\sb K\cdot c\sb K$, where $h\sp-\sb K$ denotes the relative class number of K and $c\sb K$ a rational factor that is explicitly given in terms of the ramification of K relative to ${\bbfQ}.$ \par The leading idea of the present paper is the concept of Dirichlet convolution, whose meaning for the construction of cyclotomic numbers is studied in detail. In particular, we use Dirichlet convolution to obtain two new types of cotangent numbers from the original ones. In the end, we get the following results: \par (1) Cotangent index formulas for $h\sp-\sb K$ containing rational factors that are much simpler than $c\sb K$. (2) Analogous index formulas for certain divisors of $h\sp-\sb K$ (so called ``branch class numbers''). (3) A simple transition from modified cotangent numbers to Stickelberger elements, which infers corresponding Stickelberger index formulas.
[K.Girstmair]

MSC 2000:: *11R29 Class numbers, class groups, discriminants
11R18 Cyclotomic extensions
11R20 Other abelian and metabelian extensions

Keywords: abelian number field; cotangent numbers; cyclotomic field; relative class number; ramification; Dirichlet convolution; Stickelberger elements

Citations: Zbl 0675.12002

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