Class number factors and distribution of residues
Class number factors and distribution of residues
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类别数因子和残留物分布
DOI：
10.1007/bf02940820
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
K. Girstmair
中科院分区：
文献类型：
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作者：
K. Girstmair
设gm /H是素数剩余群gm = (0 /m 0)x的因子群。我们引入了包含在ε Gm/H类中整数（1≤k≤m, (k, m) = 1）分布的重要数据的b-分割向量(b)。特别令人感兴趣的是欧氏长度ofT(b)及其分量的符号。首先，我们用ofT(b)表示了它们的阿贝尔子场的若干类数因子。这些表达式可以用作类数公式，但我们更感兴趣的是相反的问题：特别是这些类数因子onT(b)对‖T(b)‖的影响是什么？关于这种影响的许多信息包含在规范流形中。这个流形是欧氏环面族，我们将详细研究它。在某些情况下。是紧凑的，这意味着类数因素完全决定了长度ofT(b)。在其余情况下，则为：仅为‖T(b)‖提供一个下界。利用广义黎曼假设，我们对这个下界的性质有了一些认识。此外，我们表明，ofT(b)与类数因子的连接产生了关于组件ofT(b)符号的有趣事实，并且在实际计算中很有帮助。
LetGm/H be a factor group of the prime residue groupGm = (ℤ/mℤ)x. We introduce theb-division vectorT(b), which contains important data on the distribution of the integersk, 1 ≤k ≤m, (k, m) = 1, amongst the classesC ε Gm/H. Of particuar interest are the euclideanlength ofT(b) and thesigns of its components. First we express certain class number factors of abelian subfields of them-th cyclotomic field in terms ofT(b). These expressions can be used as class number formulas, but we are more interested in the converse question: What is the influence of the said class number factors onT(b), in particular, on ‖T(b)‖? Much information about this influence is contained in thenorm manifold ℒ. This manifold is a family of euclidean tori, which we study in detail. In some cases ℒ. is compact, which means that class number factors determine the length ofT(b) completely. In the remaining cases ℒ. supplies a lower bound for ‖T(b)‖ only. We obtain some insight into the quality of this lower bound by means of the Generalized Riemann Hypothesis. Moreover, we show that the connection ofT(b) with class number factors yields interesting facts about the signs of the components ofT(b) and is helpful in actual computations.