On the Value Distribution of Arithmetic Functions Original Research Article

In part II of a series of articles on the least common multiple, the central object of investigation was a particular integer-valued arithmetic functiong1(n). The most interesting problem there was the value distribution ofg1(n). We proved that the counting function card{nless-than-or-equals, slantx: g1(n)less-than-or-equals, slantd} has orderod(x) for any fixedd. A characteristic feature ofg1(n) is its so-called super-periodicity which will be discussed here. An integer-valued arithmetic functiong(n) is calledsuper-periodic, if there is a sequence (rj) of positive integers withrjgreater-or-equal, slanted2 (jgreater-or-equal, slanted2) such that, settingRkcolon, equals∏kj=1 rj,g(rRk+j)greater-or-equal, slantedg((r−1) Rk+j) for allkgreater-or-equal, slanted1, 1less-than-or-equals, slantr