An identity for a class of arithmetical functions of two variables

[摘要] For a positive integerr, letr∗denote the quotient ofrby its largest squarefree divisor(1∗=1).Recently, K. R. Johnson proved that(∗)∑d|n|C(d,r)|=r∗∏pa‖nr∗p+r(a+1)∏pa‖nr∗p|r(a(p−1)+1)   or   0according asr∗|nor not whereC(n,r)is the well known Ramanujan's sum. In this paper, using a different method, we generalize(∗)to a wide class of arithmetical functions of2variables and deduce as special cases(∗)and similar formulae for several generalizations of Ramanujan''s sum.

[学科分类] 数学（综合）