On a Generalization of Euler's 0-function

Professor Tanaka extended Euler's ƒÓ-function and obtained the formula to give the value of the function [2]. In this note, we ex-tend these further to any algebraic number field. Let _?? _ be the ring of algebraic integers of a (finite) algebraic number field. Let Sam, be the set of elements x+nt of 0/nt such that (x, m)=(x+a,m)=1 for fixed a•¸_?? _ and an (integral) ideal nt of _??_. DEFINITION. We define the generalized Euler's ƒÓ-function ƒÓa(m) to be the number of elements of Sam. LEMMA 1. Let in, m ' be ideals of _?? _ such that (m, m') =1, then we have ƒÓa(mm')=ƒÓa(m)ƒÓa(m'). PROOF. We have by the map And the result follows from the fact that if and only if (x, m)=(x+a, m)=(x, m')=(x+a, m')=1. Now let Ta,m be the set of elements x+m of _??_/m such that x2•{ax•ß0 (m). We set N,m to be the number of elements of Ta,,a