Arithmetical Properties On The Rings Of Algebraic Integers

This thesis mainly focuses on some arithmetical properties on the rings of alge-braic integers.In Chapter 1,we introduce the background and give some main results.In Chapter 2,we study a class of binary operations of arithmetical functions,which called regular convolution.In 1965,Menon proved a classical identity:for any positive integer n,(?)(a-1,n)=?(n)?(n),where ?(n)is the Euler totient function and ?(n)is the divisor function.We study the Menon's identity with the regular convolutions in residually finite Dedekind domains.See Theorem 2.3.2 and Theorem 2.3.4.In Chapter 3,we consider the action of group(Z/nZ)x on the set Z/nZ,by using Cauchy-Frobenius-Burnside lemma,we mainly obtain:let p be an odd prime and a a positive integer.Suppose that Hk a subgroup of(Z/paZ)x with index k.If k=p?u,0??<a,u|p-1,then(?)(a-1,p?)=?(p?)/k(1+k(?-?)+u·p?-1/p-1).In Chapter 4,let Ok be the ring of algebraic integers in a number field K and n a nonzero ideal of OK.Let ?f be a character modulo n with conductor(?).Then(?)N(<a+b-1)+n)xf(a)=?((?))?(n02/(?))X(n/n0)?(n/(?)),where n0|n such that n0 has the same prime ideal factors with(?)and(n0,n/n0)=OK.See Theorem 4.3.1.In Chapter 5,we consider the Ramanujan's sum in OK.Let n be a nonzero ideal of OK and m an ideal of OK.Then there exists an ideal R satisfying(R,mnD0)=1 and mn-1D0-1 R is a fractional principal ideal,where D0 is the different of OK.Let y ? be a generator of mn-1D0-1R,i.e.