# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Dirichlet convolution of cotangent numbers and relative class number formulas. (English) Zbl 0717.11048

Let n be the conductor of an absolutely abelian number field K. The “cotangent numbers” $i·cot\left(\pi k/n\right)$, $\left(k,n\right)=1$, belong to the n-th cyclotomic field. Their K-traces generate an additive subgroup ${S}_{K}$ of the ring ${𝒪}_{K}$ 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 ${𝒪}_{K}\cap i·ℝ$ equals ${h}_{K}^{-}·{c}_{K}$, 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 $ℚ·$

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: K.Girstmair
##### MSC:
 11R29 Class numbers, class groups, discriminants 11R18 Cyclotomic extensions 11R20 Other abelian and metabelian extensions
##### References:
 [1] Apostol, T. M.: Introduction to Analytic Number Theory. New York-Heidelberg-Berlin: Springer. 1976. [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. [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.