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:

    “Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
    —Nadine Gordimer (b. 1923)

    “Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
    —Walter Pater (1839–1894)

    “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)