Ramanujan Expansions Of Arithmetic Functions Of Several Variables In Algebraic Number Field

S.Ramanujan defined the following classic Ramanujan sum:cq(n)=(?)e2?ikn/q(q,n?N),(0-1)where N is the set of positive integers,and(k,q)is the greatest common divisor of k and q.In 1976,based on the results of Wintner,Delange proved that all univariate arithmetic functions defined on the integer ring Z can be expanded through to Ramanujan sums.In 2018,Tóth proved that the multivariate arithmetic function defined on Z can be expanded by Ramanujan and unitary Ramanujan sums.In 2019,Hu and Qi extended the results from the rational integer ring to polynomial rings over a finite field,and proved that a large class of arithmetic functions defined on Fq[t]can be expanded through the polynomial Ramanujan sum and the unitary polynomial Ramanujan sum,which is similar to the Fourier series expansion in classical analysis.In this thesis,we extend to above results to general number rings D and show that a large class of arithmetic functions on D can be expanded through the Ramanujan and the unitary Ramanujan sums.In addition,we also investigate the multiplicative and orthogonal relations for the Ramanujan sum and the unitary Ramanujan sum defined on D.