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.

Read more about this topic:  Dirichlet Convolution

Famous quotes containing the word definition:

    “It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
    —Jane Adams (20th century)

    “Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
    —Elaine Heffner (20th century)

    “I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)