Ein v. Staudt-Clausenscher Satz für periodische Bernoullizahlen

Letk, n be relatively prime integers, 1k<n. The numbersB _k ^m =m(i/2) ^m cot^(m–1)(k/n),m2, belong to then-th cyclotomic field. They occur as values of certain simple Dirichlet series at integral arguments and have a natural interpretation as generalized Bernoulli numbers. Therefore the arithmetic properties of these numbers arouse some interest. Ifn is not a prime power,B _k ^m /m is an algebraic integer. In the other case, i.e.n=p ^e ,p prime, the fractional part ofB _k ^m /m resp.B _k ^m is determined in this paper. It is periodic modulop–1 and exhibits a connection between these Bernoulli numbers and the Stirling numbers of the second kind.     

Nicolaus Cusanus und implizite Funktionen

Mathematische Semesterberichte - Nicolaus Cusanus’ Beiträge zur Mathematik des 15.&nbsp;Jahrhunderts betreffen hauptsächlich Kreisquadraturen. Wir besprechen seinen...     

On the construction of primitive elements in field extensions

The subject of this article are a priori constructions of primitive elements in field extensions. Consider the totality of all separable polynomialsf of degreen over a fieldK with rootsx ₁,...,x _n and prescribed Galois groupG. A vector (b ₁,...,b _n )K ⁿ is called stably primitive (forG), if, for each suchf,b ₁ x ₁+...+b _n x _n generates the splitting field off. We develop representation theoretical devices to investigate the set of stably primitive vectors geometrically. A fundamental observation is that is either very large or very small (or even empty). These two cases are illustrated by various examples. Moreover, criteria are given to decide which case holds. For a number of groups where is recognized to be small we show.     

John Napiers Trigonometrie – ein Blick zurück

Zusammenfassung  Dieser Artikel gibt einen überblick über Napiers Beitr?ge zur sph?rischen Trigonometrie, wobei die dafür notwendigen sachlichen und historischen Grundlagen weitgehend bereitgestellt werden. Da diese Beitr?ge in der Literatur trotz ihrer erheblichen Bedeutung ganz im Schatten von Napiers Logarithmen stehen, erscheint eine solche Zusammenschau gerechtfertigt. Auch kommen einige neue oder kaum bekannte Einzelheiten zur Sprache. Mathematics subject classification (2000)  01A45 · 51-03     

Continued fractions and Dedekind sums: Three-term relations and distribution

Let 0?m<N be coprime integers and T(m,N) the sum of the digits of the continued fraction expansion of m/N. We study the distribution of large values of T(m,N) (“large” means T(m,N)?Nα for some fixed exponent α when N tends to infinity), which seems to be quite similar to that of the absolute values of the corresponding Dedekind sums S(m,N). This similarity is (partially, at least) due to analogous three-term relations holding for both kinds of sums. We show, e.g.,

     

Dirichlet convolution of cotangent numbers and relative class number formulas

Letn be the conductor of an abelian number fieldK. The numbersicot (k/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.