Dirichlet Convolution - Definition

Definition

If ƒ and g are two arithmetic functions (i.e. functions from the positive integers to the complex numbers), one defines a new arithmetic function ƒ * g, the Dirichlet convolution of ƒ and g, by

\begin{align} (f*g)(n) &= \sum_{d\,\mid \,n} f(d)g\left(\frac{n}{d}\right) \\ &= \sum_{ab\,=\,n}f(a)g(b) \end{align}

where the sum extends over all positive divisors d of n, or equivalently over all pairs (a, b) of positive integers whose product is n.

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