Editing Dirichlet convolution (section)

==Properties==

The set of arithmetic functions forms a [[commutative ring]], the '''{{visible anchor|Dirichlet ring}}''', with addition given by [[pointwise addition]] and multiplication by Dirichlet convolution. The multiplicative identity is the [[unit function]] <math>\varepsilon</math> defined by <math>\varepsilon(n)=1</math> if <math>n=1</math> and <math>0</math> otherwise. The [[unit (ring theory)|unit]]s (invertible elements) of this ring are the arithmetic functions <math>f</math> with <math>f(1) \neq 0</math>.

Specifically, Dirichlet convolution is [[associativity|associative]],<ref>Proofs are in Chan, ch. 2</ref>
:<math>(f * g) * h = f * (g * h),</math>
[[distributivity|distributive]] over addition
:<math>f * (g + h) = f * g + f * h</math>,
[[commutativity|commutative]],
:<math>f * g = g * f</math>,
and has an identity element,
: <math>f * \varepsilon</math> = <math>\varepsilon * f = f</math>.
Furthermore, for each function <math>f</math> having <math>f(1) \neq 0</math>, there exists another arithmetic function <math>f^{-1}</math> satisfying <math>f * f^{-1} = \varepsilon</math>, called the '''{{visible anchor|Dirichlet inverse}}''' of <math>f</math>.

The Dirichlet convolution of two [[multiplicative function]]s is again multiplicative, and every not constantly zero multiplicative function has a Dirichlet inverse which is also multiplicative. In other words, multiplicative functions form a subgroup of the group of invertible elements of the Dirichlet ring. Beware however that the sum of two multiplicative functions is not multiplicative (since <math>(f+g)(1)=f(1)+g(1)=2 \neq 1</math>), so the subset of multiplicative functions is not a subring of the Dirichlet ring. The article on multiplicative functions lists several convolution relations among important multiplicative functions.

Another operation on arithmetic functions is pointwise multiplication:  <math>fg</math> is defined by <math>(fg)(n)=f(n)g(n)</math>. Given a [[completely multiplicative function]] <math>h</math>, pointwise multiplication by <math>h</math> distributes over Dirichlet convolution: <math>(f * g)h = (fh) * (gh)</math>.<ref>A proof can be found in [[Completely multiplicative function#Proof of distributive property|this article]].</ref> The convolution of two completely multiplicative functions is multiplicative, but not necessarily completely multiplicative.