高级Rademacher符号

IF 0.7 4区数学 Q2 MATHEMATICS

Experimental Mathematics Pub Date : 2023-07-21 DOI:10.1080/10586458.2023.2219071

W. Duke

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引用次数: 1

摘要

.定义了某些较高的Rademacher符号，这些符号给出了模组上的类函数。它们的基本性质是通过Eichler-Shimura上同调的双变量重新表述得到的。这种重新表述更好地解释了循环积分的作用，并导致了新的评估。Rademacher符号确定了实二次域的窄理想类的zeta函数的非正整数值。这一结果与Siegel的结果相当，但通过使用一个在大于1的正整数上的ζ函数值的恒等式作为为二次域定义的某些双ζ值的和，以一种新的方式证明了这一结果。

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Higher Rademacher Symbols

. Certain higher Rademacher symbols are deﬁned that give class functions on the modular group. Their basic properties are derived via a two-variable reformulation of Eichler-Shimura cohomology. This reformulation better explains the role of cycle integrals and leads to new evaluations. The Rademacher symbols determine the values at non-positive integers of the zeta function for a narrow ideal class of a real quadratic ﬁeld. This result is equivalent to one of Siegel, but is proven in a new way by using an identity for the value of such a zeta function at a positive integer greater than one as a sum of certain double zeta values deﬁned for the quadratic ﬁeld.

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来源期刊

Experimental Mathematics 数学-数学

CiteScore

1.70

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses. Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was "through systematic experimentation.") Yet this tends to be concealed by the tradition of presenting only elegant, fully developed, and rigorous results. Experimental Mathematics was founded in the belief that theory and experiment feed on each other, and that the mathematical community stands to benefit from a more complete exposure to the experimental process. The early sharing of insights increases the possibility that they will lead to theorems: An interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere. Even when the person who had the initial insight goes on to find a proof, a discussion of the heuristic process can be of help, or at least of interest, to other researchers. There is value not only in the discovery itself, but also in the road that leads to it.