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其中？是正整数集. (k, q) 是 k 和 q 的最大公因子. 1976年, 在 Wintner 的结果的基础上， Delange 证明了定义在整数环？上的单变量算术函数可以通过 Ramanujan 和加以展开. 2018 年, T′oth 证明了定义在？上的多元算术函数可以通过 Ramanujan 和与酉 Ramanujan 和加以展开. 在此基础上, 本文试图将定义在代数整数环上的多元理想函数通过 Ramanujan 和加以展开, 同时也将进一步研究代数整数环上 Ramanujan 和的乘性与正交关系等性质.One hundred years ago, Ramanujan first defined the following classic Ramanujan sum:
![\"\"](\"Edit_7f02e2ff-4ee8-4a8b-8f6f-f8e0f370b9a6.png\")
where ？ is the set of positive integers, and (k, q) is the greatest common factor of k and q. In 1976, on the basis of Wintner’s results, Delange proved that all univariate arithmetic functions defined on the integer ring ？ can be expanded by Ramanujan sum. In 2018, T′oth proved that the multivariate arithmetic function defined on ？ can be expanded by Ramanujan sum and unitary Ramanujan sum. On this basis, this paper attempts to expand the multivariate ideal function defined on D through the Ramanujan sum. At the same time, it will further study the multiplicative and orthogonal relations of the Ramanujan sum on D.