Introduction elthough they re omplex n lyti D the LEfun tions refle tn rithmeti n tureF hiri hlet9s heoremD whi h sserts th t there re infinitely m ny prime num ers in rithmeti progressionsD is one of the f mous ex mples whi h rel te n rithE meti pro lem to the study on the LEfun tionsF he l ss num er formul est E lish onne tion etween the spe ilv lues of LEfun tions nd sever l inv ri nts of the num er fields in luding the l ss num ersF he l nd of num er theory hs een enri hed sin e sw sw introdu ed his theory on the ZpEextensions of num er fields whi h is developed snn logue of simil r phenomenons in the fun tion fieldsF por prime num er pD the Zp-extension of num er field K is tower of num er fields