In this paper, we study Ramanujan sums \(c_{\mathcal {J}}({\mathcal {I}})\), where \({\mathcal {I}}\) and \({\mathcal {J}}\) are integral ideals in an arbitrary quadratic number field. In particular, the asymptotic behaviour of sums of \(c_{\mathcal {J}}({\mathcal {I}})\) over both \({\mathcal {I}}\) and \({\mathcal {J}}\) is investigated.

中文翻译：

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Ramanujan的平均大小在二次数字段上求和

在本文中，我们研究Ramanujan和\（c _ {\ mathcal {J}}（{\ mathcal {I}}）\），其中\（{\ mathcal {I}} \\）和\（{\ mathcal {J }} \）是任意二次数字段中的积分理想。特别是，\（{_ mathcal {I}}）\）和\（{\ mathcal {I}} \\）和\（{\ mathcal { J}} \）被调查。