Abstract
Letn be the conductor of an abelian number fieldK. The numbersicot (πk/n), (k, n)=1, belong to, then-th cyclotomic field; theirK-traces form an additive group whose index in the “imaginary part” of the ringO K involves the relative class numberh −K ofK. This was shown previously. In the present paperh −K is decomposed into “branch factors”, each of which is shown to be the index of an additive group of modified cotangent numbers. Put together in the right way, the said numbers yield formulas forh −K simpler than the previous ones. The different types of cotangent numbers are mutually connected by Dirichlet convolution, whose meaning in the construction of cyclotomic numbers is studied. Finally, our results are rephrased in terms of Stickelberger ideals.
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Girstmair, K. Dirichlet convolution of cotangent numbers and relative class number formulas. Monatshefte für Mathematik 110, 231–256 (1990). https://doi.org/10.1007/BF01301678
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DOI: https://doi.org/10.1007/BF01301678