Generalized Eisenstein series and modified Dedekind sums.
Generalized Eisenstein series and modified Dedekind sums.
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广义爱森斯坦级数和修正的戴德金和。
DOI：
10.1515/crll.1975.272.182
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发表时间：
1975
期刊：
影响因子：
--
通讯作者：
B. Berndt
中科院分区：
文献类型：
--
作者：
B. Berndt
J. Lewittes [7]发现了一种获得某些广义Eisenstein级数变换公式的方法。由这些公式可以导出推广经典Dedekind η-函数η（z）的函数的变换公式。然而，Lewittes的结果是如此复杂，以致于即使在η（z）的最简单情况下，通常的关于Dedekind和的变换公式也极难推导。文[3]中我们证明，通过重做Lewittes证明的最后部分，我们可以得到相当简单的一般变换公式。然后，我们证明了文献中的许多结果可以很容易地从一般定理推导出来。本文考虑一类比上述更一般的Eisenstein级数。所得到的变换公式包括前面提到的公式。我们的证明包含了对Lewittes工作的进一步简化。在变换公式中出现了含指数因子的Dedekind和的新推广。互易定理证明了这些修改的Dedekind和。
J. Lewittes [7] has discovered a method of obtaining transformation formulae for certain generalized Eisenstein series. From these formulae, transformation formulae for functions which generalize the classical Dedekind eta-function η (z) can be dedueed. However, Lewittes' results are so complieated, that even in the simplest case of η (z), the usual transformation formulae in terins of Dedekind sums are exceedingly difficult to deduce. In [3], we showed that by redoing the last parts of Lewittes' proof, we could obtain considerably simpler general transformation formulae. We then showed that many results in the literature could be easily dedueed from the general theorem. In this paper, we consider a more general class of Eisenstein series than that mentioned above. The transformation formulae obtained include the previously mentioned formulae. Our proof incorporates still further simplifications on Lewittes' work. Appearing in the transformation formulae are new generalizations of Dedekind sums involving exponential factors. Reciprocity theorems are proved for some of these modified Dedekind sums.