Girstmair, Kurt Dirichlet convolution of cotangent numbers and relative class number formulas. (English) Zbl 0717.11048 Monatsh. Math. 110, No. 3-4, 231-256 (1990). 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 \(S_ K\) of the ring \({\mathcal O}_ K\) of integers of \(K\). Previously [J. Number Theory 32, 100–110 (1989; Zbl 0675.12002)] we have shown that the group index of \(S_ K\) in \({\mathcal O}_ K\cap i\cdot \mathbb R\) equals \(h^-_ K\cdot c_ K\), where \(h^-_ K\) denotes the relative class number of \(K\) and \(c_ K\) a rational factor that is explicitly given in terms of the ramification of \(K\) relative to \(\mathbb Q\). 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: (1) Cotangent index formulas for \(h^-_ K\) containing rational factors that are much simpler than \(c_ K\).(2) Analogous index formulas for certain divisors of \(h^-_ K\) (so called “branch class numbers”).(3) A simple transition from modified cotangent numbers to Stickelberger elements, which infers corresponding Stickelberger index formulas. Reviewer: Kurt Girstmair Cited in 2 Documents MSC: 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 PDFBibTeX XMLCite \textit{K. Girstmair}, Monatsh. Math. 110, No. 3--4, 231--256 (1990; Zbl 0717.11048) Full Text: DOI EuDML Digital Library of Mathematical Functions: §24.16(iii) Other Generalizations ‣ §24.16 Generalizations ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials References: [1] Apostol, T. M.: Introduction to Analytic Number Theory. New York-Heidelberg-Berlin: Springer. 1976. · Zbl 0335.10001 [2] Girstmair, K.: Ein v. Staudt ? Clausenscher Satz f?r periodische Bernoullizahlen. Mh. Math.104, 109-118 (1987). · Zbl 0626.12001 · doi:10.1007/BF01326783 [3] Girstmair, K.: Character coordinates and annihilators of cyclotomic numbers. Manuscr. Math.59, 375-389 (1987). · Zbl 0624.12006 · doi:10.1007/BF01174800 [4] Girstmair K.: An index formula for the relative class number of an abelian number field. J. Number Th.32, 100-110 (1989). · Zbl 0675.12002 · doi:10.1016/0022-314X(89)90100-5 [5] Hasse, H.: ?ber die Klassenzahl abelscher Zahlk?rper. Berlin: Akademie-Verlag. 1952. [6] Iwasawa, K.: A class number formula for cyclotomic fields. Ann. of Math.76, 171-179 (1962). · Zbl 0125.02003 · doi:10.2307/1970270 [7] Kubert, D. S., Lang, S.: Stickelberger ideals. Math. Ann.237, 203-212 (1978). · Zbl 0379.12009 · doi:10.1007/BF01420176 [8] Leopoldt, H. W.: ?ber die Hauptordnung der ganzen Elemente eines abelschen Zahlk?rpers. J. reine angew. Math.201, 119-149 (1959). · Zbl 0098.03403 · doi:10.1515/crll.1959.201.119 [9] Lettl, G.: The ring of integers of an abelian number field. J. reine angew. Math.404, 162-170 (1990). · Zbl 0703.11060 · doi:10.1515/crll.1990.404.162 [10] Lettl, G.: Stickelberger elements and cotangent numbers. To appear. · Zbl 0757.11038 [11] Sinnott, W.: On the Stickelberger ideal and the circular units of an abelian field. Invent. Math.62, 181-234 (1980). · Zbl 0465.12001 · doi:10.1007/BF01389158 [12] Washington, L. C.: Introduction to Cyclotomic Fields. New York-Heidelberg-Berlin: Springer. 1982. · Zbl 0484.12001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.