The
Dirichlet convolution of
arithmetic functions ƒ and
g, is the arithmetic function
ƒ *
g defined by
where
d runs over all positive divisors of
n. This allows us to express certain sums over divisors more concisely and let us see their more elegant structure.
Properties
The Dirichlet convolution has some nice properties. It is
commutative
and
associative
and
distributive over addition
In fact, arithmetic functions together with * and + operations form a
ring.
Dirichlet convolution's Identity Function
The
unit function (or ` \epsilon ` function)
works like the
Kronecker delta or the
Dirac delta. Think of it as a light bulb that switches on (gives a '1') when given ` n = 1 ` as an input and turns off when n is given any other value as input. Under the operation of Dirichlet convolution (* operation), this function plays the role of the
*-
identity. In other words, for all arithmetic functions ` f `,
The constant 1 function and the function identity ` I ` do not play the same role, unlike what we would expect for ordinary multiplication and
function composition. However, they are related by the formula
which is described
here and proven
here.
Divisor Sums and the inverse of the '1' function
The divisor sum ` Df ` of ` f ` is
Thus we can think of the ` D ... ` operator as formally the same as ` 1 \mbox{*} ... ` The Inverse of ` 1 ` function under Dirichlet convolution turns out to be the Möbius function
` \mu ` as
This leads to the celebrated
Möbius Inversion Formula.
This leads to the celebrated Möbius Inversion Formula.
