{"title":"Higher Rademacher Symbols","authors":"W. Duke","doi":"10.1080/10586458.2023.2219071","DOIUrl":null,"url":null,"abstract":". 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.","PeriodicalId":50464,"journal":{"name":"Experimental Mathematics","volume":" ","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2023-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Experimental Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1080/10586458.2023.2219071","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
. 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.
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.
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