Generalizations of Dirichlet Convolution

Abstract

Let K be a complex-valued function on the set of all ordered pairs <n,d> where n is a positive integer and d is a divisor of n. If f and g are arithmetical functions, their K-convolution, f *_K g, is defined by

$$ (f{*_K}g)(n) = \sum\limits_{{d\left| n \right.}} K (n,d)f(d)g(n/d)\quad {\text{for}}\,{\text{all}}\,{\text{n}} $$