On the distribution of Ramanujan sums over number fields

For a number field $${\mathbb {K}}$$ , and integral ideals $${\mathcal {I}}$$ {J}}$$ in its ring {O}}_{{\mathbb {K}}}$$ Nowak studied the asymptotic behavior of average Ramanujan sums $$C_{{\mathcal {J}}}({{\mathcal {I}}})$$ over both . In this article, we extend investigation by establishing formulas for second moment averages quadratic cubic fields, thereby generalizing previous works Chen, Kumchev, Robles, Roy on moments rationals. Additionally, using special property certain domains, obtain results some other fields.