Arithmetic Functions and the Cross-Convvolution

The Cross - convolution product of two arithmetic functions f and g is defined as ( )( ) ( ) ( ) ( ) ( ) ( ) f g n n f x g y xy n x y , where (n) 0 for all nN . We establish properties of arithmetic function under Cross - convolution and compare our results with those of the Dirichlet convolution. Keywords - Multiplicative function, Completely multiplicative function, Dirichlet convolution, Cross – convolution